For more information, please see full course syllabus of Math Analysis

For more information, please see full course syllabus of Math Analysis

### Instantaneous Slope & Tangents (Derivatives)

- Long ago in algebra, we were introduced to the concept of
*slope*. We can think of slope as the rate of change that a line has. We define slope as the vertical change divided by the horizontal:

m = rise

. - The idea of slope makes sense for a line, because the slope is always the same, no matter where you look. However, on most functions, the slope is constantly changing. While we can't talk about a rate of change/slope for the entire function, we can look
for a way to find the
*instantaneous slope*: the slope at a point. - Another way to consider the idea of instantaneous slope is through a
*tangent line*: a line going in the "same direction" as a curve at some point, and just touching that one point. - We can find the average slope between two points on a function with

m _{average}=f(x _{2}) − f(x_{1})

_{2}− x_{1}

. - Alternatively, we can rephrase the above formula by considering the average slope between some location x and some location h distance ahead:

m _{avg}=f(x+h) − f(x)

. - Notice that as h grows smaller and smaller, our slope approximation becomes better and better. With this idea in mind, we take the limit of the above as h→ 0. Assuming the limit exists, we have the
*derivative*:

f′(x) =

lim

f(x+h) − f(x)

. *differentiation*. - There are many ways to denote the derivative. Given some y = f(x), we can denote it with any of the following:

f′(x) dx

y′ d

⎡

⎣

f(x) ⎤

⎦

- As you progress in calculus, you will quickly learn a wide variety of rules and techniques to make it much easier to find derivatives. You will very seldom, if ever, use the formal definition to find a derivative. The
__really important__idea is what a derivative represents. A derivative is a way to talk about the instantaneous slope (equivalently, rate of change) of a function at some location. No matter how many rules you learn for finding derivatives, never forget that it is, at heart, a way to talk about a function's moment-by-moment change. - One rule to make it easier to find the derivative of a function is the
*power rule*. It says that for any function f(x) = x^{n}, where n is a constant number, the derivative is

f′(x) = n ·x ^{n−1}.

### Instantaneous Slope & Tangents (Derivatives)

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro
- Introduction
- The Derivative of a Function Gives Us a Way to Talk About 'How Fast' the Function If Changing
- Instantaneous Slop
- Instantaneous Rate of Change
- Slope
- Idea of Instantaneous Slope
- Tangent to a Circle
- Tangent to a Curve
- Towards a Derivative - Average Slope
- Towards a Derivative - General Form
- Towards a Derivative - General Form, cont.
- An h Grows Smaller, Our Slope Approximation Becomes Better
- Towards a Derivative - Limits!
- Towards a Derivative - Checking Our Slope
- Definition of the Derivative
- Notation for the Derivative
- The Important Idea
- Example 1
- Example 2
- Example 3
- The Power Rule
- Example 4

- Intro 0:00
- Introduction 0:08
- The Derivative of a Function Gives Us a Way to Talk About 'How Fast' the Function If Changing
- Instantaneous Slop
- Instantaneous Rate of Change
- Slope 1:24
- The Vertical Change Divided by the Horizontal
- Idea of Instantaneous Slope 2:10
- What If We Wanted to Apply the Idea of Slope to a Non-Line?
- Tangent to a Circle 3:52
- What is the Tangent Line for a Circle?
- Tangent to a Curve 5:20
- Towards a Derivative - Average Slope 6:36
- Towards a Derivative - Average Slope, cont.
- An Approximation
- Towards a Derivative - General Form 13:18
- Towards a Derivative - General Form, cont.
- An h Grows Smaller, Our Slope Approximation Becomes Better
- Towards a Derivative - Limits! 20:04
- Towards a Derivative - Limits!, cont.
- We Want to Show the Slope at x=1
- Towards a Derivative - Checking Our Slope 23:12
- Definition of the Derivative 23:54
- Derivative: A Way to Find the Instantaneous Slope of a Function at Any Point
- Differentiation
- Notation for the Derivative 25:58
- The Derivative is a Very Important Idea In Calculus
- The Important Idea 27:34
- Why Did We Learn the Formal Definition to Find a Derivative?
- Example 1 30:50
- Example 2 36:06
- Example 3 40:24
- The Power Rule 44:16
- Makes It Easier to Find the Derivative of a Function
- Examples
- n Is Any Constant Number
- Example 4 46:26

### Math Analysis Online

### Transcription: Instantaneous Slope & Tangents (Derivatives)

*Hi--welcome back to Educator.com.*0000

*Today, we are going to talk about instantaneous slope and tangents, which are also called derivatives.*0002

*While there are many other things that we could explore in limits, we now have enough of an understanding*0008

*to move on to another major topic in calculus, the derivative.*0012

*The derivative of a function gives us a way to talk about how fast the function is changing.*0016

*It allows us to find the instantaneous slope, which is also called the instantaneous rate of change; they mean equivalent things.*0021

*And this is a new idea, which we will go over in just a moment.*0028

*At first glance, knowing a function's rate of change at any location may not seem that useful.*0031

*But actually, it tells us a massive amount of information.*0036

*It lets us easily find maximums and minimums; it lets us find increasing or decreasing intervals, and many other things.*0039

*Knowing the derivative of a function is really, really useful.*0046

*For example, if we have some function that gives the location of an object,*0049

*the derivative of that function will tell us the object's velocity, because derivative tells us the rate of change.*0054

*If we know something's location, and then we talk about the rate of change of that location,*0060

*well, the rate of change of your location is what your speed is, effectively.*0064

*That gives us velocity, since velocity is pretty much speed.*0068

*This is really useful stuff; being able to talk about derivatives of a function is really, really useful, and it forms one of the cornerstones of calculus.*0072

*Let's check it out: Long, long ago, when we first took algebra, we were introduced to the concept of slope.*0079

*We can think of slope as the rate of change that a line has, how fast it is moving, in a way.*0086

*That is, how far up does it go for going some amount horizontally?*0093

*We define slope as the vertical change, divided by the horizontal, which is also the rise, divided by the run:*0098

*rise over run, the amount that we have changed vertically, divided by the amount that we have changed horizontally.*0108

*This tells us how fast our line is changing; it tells us the rate of change, the slope, how much it moves on a moment-by-moment basis for a line.*0114

*For one step to the right, how much will we go up or down?*0123

*The idea of slope makes a lot of sense for a line, because its rate of change is always constant.*0128

*But if we wanted to apply the idea of slope to a non-line, something like, say, a parabola, for example?*0133

*The first thing to notice is that, for most functions, slope is constantly changing--*0139

*not necessarily for all functions (for a line, it isn't changing), but for anything that isn't a line, the slope won't be the same everywhere.*0144

*The rate of change for a function varies depending on what location we consider.*0150

*For example, on this one, how fast it is changing is totally different in this area; it is totally different from this area and totally different from this area.*0154

*Each of those three areas is going in a very different rate of change.*0162

*The way it is moving there is very different.*0166

*In this area over here, it is mainly going down; in this area over here, it is mainly just going horizontally; and in this area over here, it is mainly going vertically.*0168

*There is always some horizontal motion in this case, but it ends up changing how it is moving.*0178

*So, it doesn't have a constant slope; its slope is changing for these things.*0183

*We want to have some way of being able to talk about what the slope is at this place.*0187

*What does it change to in this moment?*0191

*So, we can't talk about a rate of change (slope) for the entire function, but we can look for a way to find the instantaneous slope.*0194

*What is the slope at some specific point?*0201

*Here, it is changing at this moment, at some current speed; it is going like this here.*0204

*But here, it is going like this; or here, it is going like this; and here, it is going like this.*0210

*We end up getting these different ways of being able to talk about how it is moving in this moment.*0219

*Where is it going from one spot to the next spot--how is it changing at that place?*0224

*Another way to work towards this idea of instantaneous slope at a point is through the notion of a tangent line.*0229

*Now, let's have just a quick break from this: there is a slight relationship between the trigonometric function "tangent" and the "tangent" line of something.*0236

*But it is really not worth getting into; it is not going to really help us understand things, and the connection is only really tenuous.*0245

*For now, let's just pretend that they are two totally different ideas, and that they just happen to have the same name.*0251

*Tangent, when we talk about taking the tangent of some angle, and when we talk about the tangent line on a curve--*0258

*they are completely unrelated, at least as far as we are concerned right now.*0264

*It is easier to think about it that way.*0268

*Back to what we were talking about: for a circle, the tangent line to a point on the circle*0270

*is a line that passes through the point, but intersects no other part of the circle.*0274

*Consider this point right here on the circle: the tangent line will go through that point, but it will intersect no other part.*0278

*We look at it, and we see that we get this right here.*0289

*See how it barely touches--it just touches, feather-like, that one point on the circle; but it touches nothing else.*0293

*I want you to notice how the tangent line is basically the instantaneous slope of the curve at that point.*0300

*If we look in this region, right here, at our point, that is how much the curve is changing at that moment.*0306

*So, it is as if the tangent line is going the same direction as the curve in that one location.*0313

*We can take this idea of a tangent line and expand it to any curve.*0320

*If we have some function f(x), when we draw its graph, we just have a curve on our paper.*0323

*Now, we can consider some point on the graph and try to find a tangent line at that point.*0329

*So, say we have some point, like right here, and we want to do the same thing*0333

*of just barely feather-like touching that one location and going in the same direction as the graph is going.*0337

*It will pass through that one place, but just barely going in the same direction as the curve is in that moment.*0342

*We look at that, and we see how that has the instantaneous slope.*0350

*Notice: the tangent line is the instantaneous slope of the curve at that point.*0354

*This curve right here has what the slope is at this one place.*0359

*But if we go to some other place, we would end up having a totally different tangent line for these different points.*0365

*If a tangent line were to pass through these different points, it would have a totally different slope.*0371

*The tangent line is going in the same direction as the curve is at that moment, at that single point.*0375

*At a different point, it might end up having a totally different tangent line, having a totally different slope.*0380

*So, how can we find this instantaneous slope--what can we do to work towards this slope at some point?*0385

*Well, let's say we wanted to find the specific instantaneous slope for the function f(x) = x ^{2} + 1 at a horizontal location of x = 1.*0392

*Here is the point that we are trying to find the instantaneous slope of.*0401

*And notice that we are currently at the horizontal location x = 1.*0404

*How could we do this--what could we do to approach it?*0409

*Well, long ago, when we talked about the properties of functions,*0411

*we noticed how we could talk about the average slope between two horizontal locations,*0414

*x _{1} and x_{2}, on a function, with this formula here.*0419

*The average slope between these two horizontal locations, x _{2} and x_{1}, is f(x_{2}) - f(x_{1})/(x_{2} - x_{1}.*0423

*Well, why is that? Well, if we have some other point that is at some x _{2}, what height will that end up being at?*0433

*Well, that will end up being at a height of...to find the height, we just evaluate the horizontal location.*0440

*Your input is your horizontal location, and your output is your vertical location.*0444

*So, that would come out to be f(x _{2}); what input did we have?*0448

*Well, we put in x _{1}, so we would have an output of f(x_{1}).*0451

*So, our top, the change, is going to be the difference: f(x _{2}) - f(x_{1}),*0456

*because we went to f(x _{2}), and we came from f(x_{1}).*0466

*The difference in our ending and our starting is f(x _{2}) - f(x_{1}).*0470

*So, that tells us the top part of our average slope formula.*0474

*The bottom part of it: well, if we go from x _{1} to x_{2}, then that means that our horizontal motion is going to be x_{2} - x_{1}.*0478

*If we go to 10 from 2, we have traveled 8, 10 minus 2; and so, that is why we divide by x _{2} - x_{1}.*0487

*It is the rise (how much we have changed vertically), divided by how much we have changed horizontally.*0495

*So, we have our function, f(x) = x ^{2} + 1; and we want to know what the slope at x = 1 is.*0501

*What is the value? We have our slope average formula, m _{avg} = f(x_{2}) - f(x_{1}) over (x_{2} - x_{1}).*0506

*Now, we want to know the slope at x = 1; so we can use this average slope formula to give us approximations.*0515

*By using the average slope formula, we can get approximations for what the slope is near x = 1, for finding our instantaneous slope at x = 1.*0523

*So, since we want to find it near x = 1 (we want to find it specifically at x = 1),*0535

*let's set the first thing that we will use in our average slope formula as x _{1} = 1; so we establish this as being x_{1}.*0540

*For our first approximation, let's use a horizontal location that happens to be two units forward.*0549

*So, we will have our second horizontal location be moved two forward; we go forward one, forward two; and that makes 3 x _{2}.*0554

*Or, we could have x _{2} = x_{1} + 2; and since we are using x_{1} = 1, we have 1 + 2, which comes out to be 3.*0561

*So, we have x _{1} and x_{2}; we are looking to find the average slope between these two points.*0573

*All right, our function is f(x) = x ^{2} + 1; we are looking for our instantaneous slope at x = 1.*0579

*We are working towards that by approximations right now.*0586

*And our average slope formula is f(x _{2}) - f(x_{1}), over (x_{2} - x_{1}).*0588

*We know that x _{1} = 1, because that is the point that we are interested in finding the slope for.*0594

*So, we will just set that as sort of a starting place to work from.*0599

*And we decided, for our first one, to go with x _{2} = x_{1} + 2, which was 3.*0602

*Using our formula, we have m _{avg} = f(x_{2}) - f(x_{1}) over (x_{2} - x_{1}).*0607

*x _{2} is 3; so if we plug that into f(x), then we have f(x_{2}) = 3; so f(x_{2}) is f(3).*0613

*f(3) would come out to be 3 ^{2} + 1; 3^{2} + 1 gets us 10.*0623

*For f(x _{1}), f(x_{1}) would be f(1); f(1) would be 1^{2} + 1, so that gets us 2.*0629

*So, it is 10 - 2 on the top, and then x _{2} (3) minus x_{1} (1), so 3 - 1 on the bottom.*0638

*We simplify the top; we get 10 - 2, which becomes 8; 3 - 1 becomes 2; 8 divided by 2 gets us 4.*0646

*And if we draw it in, we end up getting this purple line right here: that is the slope if we set it equal to that average slope.*0652

*It ends up passing through those two points; so we have an average slope of 4 between those two horizontal locations.*0660

*OK, that is not a bad start; it is far from perfect--we can clearly see that this line right here is not, in fact, the tangent line.*0666

*It is not the tangent line; it doesn't pass perfectly against that point; it doesn't just barely, feather-like, touch that one point.*0676

*It isn't going in the same direction, but it does give us an approximation; it is a start.*0682

*How can we make this approximation better?*0688

*Well, we could probably think, "Well, the issue here is the fact that x _{2} is too far away."*0690

*We want x _{2} to be closer; so we can improve it by bringing x_{2} closer to x_{1}.*0694

*This time, let's go only one away; we will do x _{1} + 1, so that it is only one distance forward.*0701

*So, we will now have x _{1} + 1, or 2; since we are starting at 1, 1 + 1 gets us 2; now, we are only one horizontal distance away.*0706

*It is still the same function, x ^{2} + 1; we are still looking for x = 1; it is still the same slope average formula.*0716

*x _{1} is still equal to 1, but now x_{2} is going to be one forward from 1, so that is 2 for our x_{2}.*0723

*We plug that into our average slope formula; f(x _{2}) is f(2) in this case, because that is our x_{2} at this place.*0730

*So, f(x _{2}) would be 2^{2} + 1; 4 + 1 gets us 5;*0738

*minus f(x _{1}): f(x_{1}) is still 1^{2} + 1, so that still gets us - 2.*0743

*Our bottom is now x _{2} - x_{1}; our new x_{2} is 2, in this case;*0749

*2 - 1 simplifies to...5 - 2 on top becomes 3; 2 - 1 on the bottom becomes 1.*0753

*And so, we get a slope of 3; and if we graph that, we end up getting this line right here.*0759

*All right, nice--we are getting better; it is still not perfect, but the approximation is improving as we bring x _{2} closer to x_{1}.*0766

*So, as we bring our x _{2} closer and closer and closer, we are going to end up getting better and better approximations.*0774

*What we want to do is bring it really close.*0780

*Before we can bring x _{2} really close, though, we need to think about what we are doing in general,*0783

*so that we can figure out an easy way to formulate talking about bringing x _{2} really, really close.*0787

*So, let's talk about what we have been doing, in general.*0793

*We have our average slope formula; m _{avg} is equal to f(x_{2}) - f(x_{1}), divided by (x_{2} - x_{1}).*0795

*That is the output of our second point, minus the output of our first point, divided by the horizontal location difference of our second and first points.*0802

*So, what we did first was (since we are looking to use this m _{avg} formula): we set x_{1} at some value.*0810

*In this case, we set it at the point that we are interested in; we wanted to find the slope of x = 1, so we set our first point,*0815

*our first horizontal location, as x _{1} = 1; the first horizontal location is 1, because we want to find out about that slope.*0821

*Then, from there, we set x _{2} some distance away from it.*0828

*The very first time, we set it 2 distance away; so 1 + 2 became 3.*0834

*The second time we did this, we had 1 + 1 (a horizontal distance of 1); 1 + 1 became 2.*0839

*So, we want to bring x _{2} closer and closer by putting less and less distance.*0844

*What we really care about isn't so much the second point, but the distance to the second point.*0848

*There are two different ways of looking at this.*0853

*So, let's call this distance something; we will call it h.*0854

*What we want to do is bring this h smaller and smaller and smaller.*0857

*We want to bring x _{2} closer and closer, so we want to make the distance between our point that we care about,*0861

*and the point that we are referencing against for our average slope, to become closer and closer and closer.*0867

*So, if we are calling this distance h, then we can say that x _{2} is equal to x_{1}, our starting place, plus the distance away, h.*0872

*So, x _{2} = x_{1} + h; with this in mind, we can now rewrite our average slope formula in terms of x_{1}.*0880

*We started with m _{avg} = f(x_{2}) - f(x_{1}), over (x_{2} - x_{1}).*0888

*But now, we have this new way of writing x _{2}: x_{2} is equal to x_{1} plus the distance forward, h.*0893

*So, we can rewrite the formula in terms of x _{1} and the horizontal distance, h, to be able to create a new formula--*0898

*not a new formula, so much as a restatement of the old formula--but a new way of looking at it.*0906

*So, we plug in x _{2} here; it now becomes x_{1} + h; so we have x_{1} + h...*0911

*f(x _{1}) is still the same; so f(x_{1} + h) - f(x_{1}); x_{2} now becomes x_{1} + h, minus...still x_{1}.*0917

*On the bottom, we have x _{1} here and -x_{1} here; so we can cancel them out.*0928

*And so, we are left with just h here; on the top, though, we can't cancel anything out,*0933

*because f(x _{1} + h) and f(x_{1})...we don't know how they compare until we apply some specific function to them.*0937

*So, we are going to have to use the function before we can cancel anything out.*0944

*We can't cancel out the x _{1} part inside of the function,*0947

*because we have to see how h interacts with x _{1} before we can cancel anything out.*0950

*OK, one last thing to notice: at this point, the only thing showing up is x _{1}.*0955

*Only x _{1} shows up in this formula: we have x_{1} here, x_{1} here, x_{1} here, x_{1} here, x_{1} here, x_{1} here.*0962

*But no longer x _{2}: we don't have to care about it anymore, because instead we are thinking in terms of this distance h.*0969

*That is how we swapped to x _{1} + h.*0976

*So, if we don't really care about x _{1} versus x_{2}, then we can just rename x_{1} as simply x,*0978

*because at heart, we are all lazy; and it is easy to write out x, compared to x _{1}--just one less thing to write.*0985

*So, we can now write our average slope as: the average slope is equal to f(x + h) - f(x), divided by h.*0992

*All right, going back to our thing, we have that our function is x ^{2} + 1 (back to our specific example);*1001

*and we want to find out what the slope is at a specific value of x = 1.*1006

*So, for any h, for any distance away from our location x = 1, the average slope is going to be equal to f(x + h) - f(x), all divided by h.*1010

*Let's see how that is the case: well, if we have x here, and we have x + h here,*1024

*then the distance forward that we have gone is x + h - x, or simply x.*1030

*So, that is why we are dividing by h, because we divide by the run; we divide by the horizontal change.*1034

*And if we want to look at the two heights, well, the height that we will end at will be f(x + h).*1038

*If we plug x + h in to get an output, it is going to be f acting on (x + h).*1043

*What is the first one? Our first location is going to be plugging in x, so that would be f(x), f acting on our input of x (so f(x)).*1048

*What is the distance that we end up having? Well, that will be f(x + h) - f(x).*1058

*And so, that is why we end up having f(x + h) - f(x) on the top.*1062

*So, for any distance h that we end up going out, that tells us what the average slope is*1067

*between our horizontal location x and the h that we end up choosing--whatever distance we end up wanting to use.*1073

*So, as we make our h smaller and smaller, we will be able to get better and better approximations.*1080

*We want to see why we end up getting better and better approximations.*1084

*Notice: let's look at a couple of different points; we choose different points, and we end up running lines through these different slopes.*1086

*We end up seeing that we get closer and closer to the actual tangent we want.*1098

*We are getting closer and closer to the tangent we want.*1107

*So, as we bring our h closer and closer, as we make our h shorter and shorter and shorter, we get better and better approximations.*1110

*So, as h grows smaller and smaller, our slope approximation becomes better and better.*1119

*Now, what would be great is if we could somehow set h equal to 0.*1124

*We want to have the smallest amount of space we can possibly have; the less distance we have, the better our approximation.*1129

*But if we were to set h equal to 0, then we would be dividing by h.*1135

*It is f(x + h) - f(x), divided by h; so if h equals 0, we would divide by 0.*1138

*We can't divide by 0, because that doesn't make sense; it is not defined.*1145

*So, what we want is...if only there was some way that we could somehow have the same effect as dividing by 0,*1149

*but not have that issue where we are actually causing it to divide by 0.*1155

*If only we could look at what it was going to become the instant before it ended up breaking...*1158

*Limits! That is what that whole thing that we were studying with limits was about!*1163

*Limits give us a way to talk about what it will become before it breaks--what it is going to become just before it ends up dissolving and not actually making sense.*1166

*So, we want that infinitesimally small (as h gets really, really, really, really close to 0) thing that ends up happening.*1176

*What we are looking for is the limit as h goes to 0.*1182

*As this becomes really, really close to actually being on top of that point, what value do we end up getting out?*1185

*That is going to be the best sense of what the instantaneous slope is.*1192

*By getting it really, really, really close, we will be able to get our best idea of what the slope is at that exact place.*1196

*All right, with this idea in mind, let's take the limit as h goes to 0 of our average slope formula for f(x).*1202

*So, our average slope formula is m _{avg} = [f(x + h) - f(x)]/h.*1209

*So, what we do is take the limit as h goes to 0, because the limit as h gets smaller and smaller...*1214

*our average slope will give us a better and better approximation.*1219

*As h goes to 0, as h becomes infinitesimally close to it, we will end up getting the best possible approximation.*1222

*The limit just before the instant it touches--that is the best possible approximation we can get for what the slope is at that place.*1228

*So, at this point, we can now plug in f(x) = x ^{2} + 1 into our specific f(x) up here*1235

*and start trying to work towards what the formula is for what the slope will be at that place, at our location x.*1241

*So, if we have f(x + h), and we are plugging it into f(x) = x ^{2} + 1,*1248

*we are plugging in x + h into something squared plus 1; so we get (x + h) ^{2} + 1,*1254

*because that is what our function does; so it is (x + h) ^{2} + 1 for our first portion;*1261

*and then minus...when we plug in just x, we end up just getting x ^{2} + 1, so - (x^{2} + 1).*1266

*And the bottom is just h, because we don't have anything to affect the bottom yet.*1273

*Limit as h goes to 0...well, we can expand (x + h) ^{2}: (x + h)^{2} becomes x^{2} + 2xh + h^{2}.*1276

*The +1 still remains; and -(x ^{2} + 1) is -x^{2} - 1.*1284

*At this point, we see that we have positive x ^{2} and negative x^{2}, so they can cancel each other.*1291

*We have +1 and -1, so they can cancel each other.*1296

*And we are left with 2xh + h ^{2} on top, all divided by h:*1299

*the limit as h goes to 0 of 2xh + h ^{2}, all divided by h.*1304

*Great; if we had just plugged in 0 initially, it would end up breaking; we would have 0/0, so we don't end up getting anything out of it.*1309

*But at this point, we can now cancel things; that is one of the things we talked about when we wanted to evaluate limits.*1315

*Remember the lesson Finding Limits: how do we find these limits?*1319

*We get them to a point where we can cancel stuff, so we can see what is going on.*1322

*So, at this point, we can cancel; we have 2xh + h ^{2}, over h, in our limit;*1326

*so we see that we can cancel this h; that will cancel the h here; and over here, it will cancel the squared and turn it to just h to the 1.*1331

*So, at this point, we have it simplified to the limit as h goes to 0 of 2x + h.*1337

*And as h goes to 0, 2x won't be affected; but the h will end up canceling out as it just drops down to 0, and we will be left with 2x.*1341

*Now, what point did we care about? We cared about the horizontal location.*1350

*We wanted to show slope at x = 1; so we now have this nice formula to find out what the slope is at some horizontal location.*1354

*So, we can plug in our x = 1, and we have the instantaneous slope when the horizontal location is x = 1.*1363

*We know what the slope of f(x) is at the single moment, that single horizontal location, of x = 1.*1370

*2 times 1...we plug in our value for our x; 2 times 1 comes out to be 2; we have found what the slope is.*1376

*Let's check--let's see it graphically: if we check this against the graph, we see*1386

*that a slope of 2 at the horizontal location x = 1 produces a perfect tangent line.*1391

*This slope of 2 at the horizontal location x = 1, right here, produces a perfect tangent line to the curve at that point.*1396

*If we draw a line that goes through that point, that has this slope of 2, we end up seeing that it does exactly what we are looking for.*1404

*It has this bare feather-like touch, just barely on that curve; it just barely touches that one point,*1413

*and it goes off in the direction that the curve has at that one instant, at that one horizontal location, at that one point; cool.*1419

*So, this leads us to define the idea of a derivative; we can do what we just did here, for this one specific function, in general.*1429

*We define the derivative: the derivative is a way to find the instantaneous slope of a function at any point.*1435

*The derivative of the function f at some horizontal location x is f prime of x (we read this f with this little tick mark as "f prime of x"),*1443

*and it is the limit as h goes to 0 of f(x + h) - f(x), all divided by h.*1454

*So, our average slope gives us a better and better sense of what is happening at that instant at that horizontal location x.*1462

*As our h shrinks down, we get a better and better sense of what is going on from this average slope thing right here.*1469

*And since we have a better and better sense, the best sense that we will have is the instant before it actually ends up breaking on us at h = 0.*1476

*So, we take the limit as h goes to 0.*1482

*Now, of course, this limit has to exist; if the limit doesn't exist, then the derivative doesn't end up working out.*1484

*But as long as the limit does exist, we manage being able to find out what that instantaneous slope is.*1489

*We call the process of taking a derivative differentiation; this is called differentiation--we take the derivative through differentiation.*1494

*You can differentiate a function to get its derivative.*1501

*And when we write it out, it is just this little tick mark right here, f'(x).*1505

*You just put this little tick mark right next to your letter that is the letter of the function; and that says the derivative of that function.*1511

*Once we have the derivative, f'(x), for some function f(x), we can find the instantaneous slope*1518

*at some specific horizontal location x = a by simply plugging it into our general derivative formula, f'(x).*1524

*If we want to know the instantaneous slope at some horizontal location a, we just plug it into f'(x), and we have f'(a),*1532

*just like we did before--we figured out that, in general, for x ^{2} + 1, f prime became 2x;*1540

*and then we wanted to know what it was at the specific horizontal location of 1.*1546

*So, we took f'(1); we plugged in 1 for our x, and we got simply 2 as the instantaneous slope at that location.*1550

*The derivative is a very, very important idea in calculus; it makes one of the absolute cornerstones that calculus is built upon.*1558

*And so, there end up being a number of different ways to denote it.*1565

*Given some y = f(x), that is, some function f(x)--or we could talk about it as the vertical location y--*1569

*we can denote the derivative with any of the following.*1577

*We can talk about the derivative with any of the following symbols:*1579

*f'(x), dx/dy, y', d/dx of f of x...and there are even some other ones.*1582

*In this course, we will end up using f'(x) for the limited period of time that we actually talk about derivatives.*1593

*But these other ones will end up being used occasionally, as well.*1598

*And there are reasons why they end up being used; they actually make sense in calculus.*1602

*We don't quite have time to talk about it right now, but as you work through calculus,*1605

*as you study calculus, see if you can start to understand why we are talking about it as dx/dy.*1608

*It has to do with these ideas of infinitesimals; but I will leave that for you, working in your calculus course.*1613

*All right, the important thing to know is that, while there are all of these different ways to talk about it--*1618

*we have just 4 right here, but there are even more, occasionally (but you will only end up experiencing really...*1622

*these two right here are really likely the most common ones you will end up seeing,*1627

*and this is the only one we will use in this course), they all do the same thing.*1631

*They represent some function; they represent the derivative that tells us the instantaneous slope for a horizontal location.*1636

*We plug in a horizontal location, and it tells us what the instantaneous slope is, what the slope is,*1643

*what the rate of change is at that one specific horizontal location.*1648

*The important idea of all of this that I really want you to take away (we will work to it) is that,*1654

*as you progress in calculus, you are quickly going to learn a wide variety of rules.*1660

*You are going to learn a lot of different rules, a lot of different techniques, that will make it really easier (much easier) to find derivatives.*1664

*Things that at first seem complicated will actually end up becoming pretty easy, as you learn rules and get used to using rules in calculus.*1671

*And in actuality, you will very seldom, if ever, use that formal definition that we just saw--*1677

*that limit as h goes to 0 of f(x + h) minus f(x), over h.*1684

*That thing won't actually end up getting used a lot; we will mainly end up using these rules and techniques that you will learn as you go through calculus.*1688

*So, if that is the case--if we end up not really using it that much--why did we learn it? What was the point of learning it?*1695

*It is because the important part, the reason why we are talking about all of this,*1700

*is to give you a sense of what the derivative represents before you get to calculus.*1703

*What we really want to take away from this is that the derivative is a way to talk about instantaneous slope.*1707

*It is a way to talk about how something is changing, and equivalent to instantaneous slope is the instantaneous rate of change.*1714

*How is the function changing in this one moment, at this one point, at this one horizontal location?*1720

*How is it changing--what is the slope right there of the function?*1726

*It is what it is right there in this function at some specific location; that is the idea of a derivative.*1729

*So, as you learn these rules, no matter how many rules you learn for finding derivatives,*1735

*never forget that, at heart, what a derivative is about is a way to talk about a function's moment-by-moment change.*1740

*It is this idea of how the function is changing right here, right now.*1751

*What is the slope at this specific place?*1755

*You will end up learning a lot of rules; you will end up learning a lot of techniques.*1758

*And it is easy to end up getting tunnel vision and focusing only on the rules and techniques and getting the right numbers out, getting the right symbols out.*1761

*But even as you are ending up working on this, try to keep that broad idea of what you are thinking about.*1767

*It is how the thing is changing--how your function is changing, on the whole.*1772

*You will have to understand how to get those correct values, how to get those correct symbols, when you differentiate--when you take the derivative.*1776

*But if you forget that the idea of all of this is to talk about the rate of change, you are missing the most important part.*1783

*The most important part that makes all of this actually have meaning--to be useful--is this idea of how the function is changing here and now.*1789

*That is why we care about the derivative--not just so that we can churn out numbers;*1796

*not just so that we can churn out symbols; but so that we can talk about how this thing is changing right here and right now.*1800

*And by understanding that the derivative represents how this thing is changing right here and right now,*1806

*and thinking about it in those terms, you will be able to understand all of the larger ideas that we get out of a derivative,*1810

*of what the derivative represents, and all of the interesting things that a derivative tells us about a function.*1816

*If you just try to memorize the techniques and rules, and that is all you focus on,*1821

*you won't have a good sense for what you are doing, and it will become very difficult in calculus.*1824

*But if you keep this idea in mind of what it represents, it will be easy to understand how things fit together.*1828

*It will make things a lot more comfortable and make things make a lot more sense.*1833

*So, the important part of all of this, that I really want you to take away, is to keep the derivative in mind*1836

*as a way of talking about how the function is changing at some specific location: what is its slope right there?*1841

*All right, we are ready for some examples.*1848

*Let f(x) = x ^{3}, and consider the location x = -2.*1850

*We have some function x ^{3}, and we are considering the location x = -2.*1855

*Approximate the slope (that is the instantaneous slope), using our slope average function f(x + h) - f(x) divided by h, and the following values for h.*1859

*For our first one, h = 2: if our x is at -2, then our x is going to be at -2 for h = 2.*1868

*x + h, this portion right here, will be equal to...well, if it is -2 for x, and h is 2, then 2 + -2 comes out to be...*1877

*well, let's write it the other way around: x + h, so -2 + 2.*1889

*-2 + 2 will come out to be 0; so now, let's plug it into our average slope formula.*1894

*The average slope is equal to f(x + h) (that was 0), minus f(x) (that is -2, the point we are concerned with),*1900

*divided by h, the distance we are going out (that is 2).*1908

*We start working this out; the average slope, f(0)...well, if it is x ^{3}, that is 0^{3} - (-2)^{3}, all divided by 2.*1911

*That comes out to be 0 minus...-2 cubed is -8...over 2; 0 - -8 becomes +8, so we have 8/2, which equals 4.*1924

*So, this first approximation at h = 2 is: we end up getting an average slope of 4.*1937

*The next one: h = 1--so once again, our first place will be x = -2, and then we are working from there.*1944

*x = -2: so x + h will be equal to -2 + 1 (we are going one distance out from -2), which simplifies to -1.*1950

*So, our average slope between -2 and -1 is going to be f(x + h), so f(-1), minus f(-2), over positive 2.*1960

*What is f(-1)? Well, that is going to end up being -1 cubed...we already worked out that - f(-2) becomes -(-2) ^{3},*1976

*which means - -8, which became just +8; so we can just write that as + 8 right now, skipping to that part...divided by 2.*1987

*-1 cubed becomes -1 times -1 times -1, or just -1, plus 8...*1995

*Oh, I'm sorry; it is not divided by 2; I'm sorry about that; our h is 1--that is what we are using as our h.*1999

*I accidentally got stuck on h = 2; we are dividing by 1, because that is the distance out: 1.*2005

*-1 cubed, plus 8, is -1 + 8; we have 7/1, which equals an average slope of 7 between x = -2 and going a distance of 1 out.*2011

*Our final one: h = 0.1: our h is still equal to the same starting location, -2, but now we are going out to the very tiny x + h = 2 + 0.1, which gets us 1.9.*2024

*So, it is just a tiny little bit forward now.*2038

*Our average slope is going to be f(-2)...sorry, not -2; we do x + h first, and this should be -1.9, because it is -2 + 0.1.*2040

*So, x + h is f(-1.9), minus f(-2), all divided by our h; that is 0.1.*2050

*f(-1.9) is going to be -1.9 cubed; we already figured out that minus -2 cubed becomes +8; divide by 0.1.*2062

*Negative -1.9, cubed, becomes negative 6.859, plus 8, divided by 0.1.*2074

*So that simplifies, up top, to 1.141 divided by 0.1.*2084

*We divide by 0.1, and that moves the decimal over, and we get 11.41; great.*2091

*And so, that gives us our final average slope of h = the tiny distance of 0.1.*2100

*So, notice: at the very large distance, we got 4; at h = 2, we got 4.*2105

*At h = 1, we got 7; at h = 0.1, we got 11.41; we are slowly approaching some specific value for what it is.*2110

*What we are trying to figure out, if we were to draw a graph here, is:*2117

*here is x ^{3}; what is the slope at x = -2?*2123

*What is the slope that goes through this one place?--that is what we are trying to approximate.*2133

*What is the m of this tangent line, of that instantaneous slope there?*2138

*The best that we have gotten so far, when we plugged in this fairly small value of 0.1, was 11.41.*2143

*If we wanted to get more accurate values from this numerical way of figuring out*2148

*what the slope starts to work towards, we can use just smaller and smaller h: 0.01, 0.000001...*2152

*And we will be able to get better and better, more accurate, results.*2158

*However, if we want to get it perfectly, we have to use the derivative--a problem about the derivative!--*2161

*for f(x) = x ^{3}, then evaluating f'(-2).*2165

*And so, once again, f'(-2) tells us the instantaneous slope of our graph at this value, -2.*2169

*So, figuring out f(-2) will tell us the slope; the slope of this is going to be this value that we get from f'(-2).*2179

*All right, so how do we figure out f prime?*2189

*Well, remember: f'(x) is equal to the limit as h goes to 0 of f(x + h) minus f(x), all over h.*2191

*In this case, our f(x) is equal to x ^{3}, so we can swap this out for the limit as h goes to 0 of f(x + h)...*2205

*so f(x + h) becomes (x + h) ^{3}--we are plugging in (x + h) into how it works over here,*2219

*so (x + h) ^{3}, minus...now we are just plugging in x, so simply x^{3}, all over h.*2228

*What is (x + h) ^{3}? Let's just work that out in a quick sidebar.*2235

*(x + h) ^{3}: we can write that as (x + h)(x + h)^{2}; that becomes x^{2} + 2xh + h^{2}...*2239

*(x + h) times x ^{2} + 2xh + h^{2}...we get x^{3};*2253

*x times 2xh is 2x ^{2}h; h times x^{2} is 1hx^{2}, or 1x^{2}h.*2258

*So, we get + 3x ^{2}h; then, x times h^{2} is 1xh^{2}; h times 2xh is 2xh^{2};*2263

*so we get a total of 3xh ^{2}, plus h^{3}; h^{2} times h gets us h^{3}.*2274

*Notice that we can also get that through the binomial expansion, if you remember.*2280

*If you recently worked on the binomial expansion, you might recognize that.*2285

*All right, so at that point, we can plug back into our limit, limit as h goes to 0...*2288

*we swap out (x + h) ^{3} for x^{3} + 3x^{2}h + 3xh^{2} + h^{3} - x^{3}, all divided by h.*2293

*At this point, we see that we have a positive x ^{3} and a negative x^{3}; they knock each other out.*2308

*Now, we see that everything has an h; great.*2313

*Currently, if we were to just plug in h goes to 0, we would get 0 + 0 + 0, divided by 0; 0/0...we can't do that.*2316

*But we can knock out the dividing by 0, because everything up top now has a factor of h in it.*2324

*So, we can knock out this h here; that knocks the h out here, turns this h ^{2} into an h^{1},*2329

*turns this h ^{3} into an h^{2}...and we now have the limit as h goes to 0 of 3x^{2} + 3xh + h^{2}.*2334

*The limit as h goes to 0...well, that is going to cause 3xh to just turn into 0.*2349

*As h goes to 0, it will crush the x; as h ^{2} goes to 0, it will crush h^{2} to 0.*2355

*So, as h goes to 0, any h ^{2} will crush h^{2} to 0; and that leaves us with just 3x^{2}; great.*2361

*So, f'(x) is equal to 3x ^{2}.*2368

*So now, we were asked to find what is the specific derivative at f'(-2).*2374

*f'(-2) = what? Well, f'(x) = 3x ^{2}, so f'(-2) will be 3(-2)^{2}.*2379

*3 times -2 squared...that becomes 4, so we have 3 times 4; and we get 12.*2391

*The instantaneous slope of the point, of the horizontal location -2, is a slope of 12,*2398

*which, if you can remember from that last example that we were just working to... when we had .1, we managed to get to 11.41.*2404

*As we were making our h smaller and smaller, we were slowly working our way*2411

*to this perfectly instantaneous slope of 12, working our way to the derivative at -2; cool.*2414

*The next one: Find the derivative of f(x) = 1/x.*2421

*The same basic method works here: limit as h goes to 0 of...I'm sorry, f prime, the derivative...*2425

*f'(x) is going to be equal to the limit as h goes to of f(x + h) - f(x), all over h.*2433

*So, we start working this out: limit as h goes to 0 of f(x + h) - f(x)...*2442

*Well, x + h...we will plug into our formula f(x) = 1/x, so f(thing) = 1/thing.*2446

*So, if we plug in x + h for our thing, we have 1/(x + h), minus f(x) (is simply going to be 1/x), all divided by h.*2453

*Oh, I forgot one important part: it equals the limit as h goes to 0 of that stuff.*2463

*We have to keep up that limit, because it is very important.*2468

*Limit as h goes to 0 of all of this stuff: now, from when we worked with finding limits,*2471

*well, we have fractions on top of and inside of a fraction...*2476

*Well, what we want is less fractions; we don't want fractions in fractions, so how can we get rid of the fractions up top?*2479

*Well, notice: if we multiply by the top, we can cancel out those fractions with x times x + h.*2484

*That will cancel out the right fraction and the left fraction on the top.*2490

*We will be left with some stuff...but we have to multiply the top and the bottom of any fraction by the same thing; otherwise, we just have wishful thinking.*2494

*x times (x + h) on the top and the bottom: we have...this is equal to the limit as h goes to 0 of 1/(x + h) times (x + h)...*2501

*well, that (x + h) here will cancel out the (x + h) here, and we will be left with just x.*2511

*We get x, minus quantity...here, the x cancels out the x here, and we are going to be left with x + h, minus (x + h), all over...*2517

*well, we can't cancel out anything on the bottom; we just have h.*2528

*And if you remember from our finding limits, it is actually going to behoove us to be able to keep the h there,*2531

*because our goal, since we are going to have h go to 0...we can't divide by 0;*2535

*so we need to somehow get that h at the bottom to cancel out and be canceled out.*2539

*Otherwise, our limit will end up getting mixed up by that h still being there.*2543

*So, h times x times x + h is on the bottom.*2547

*At this point, we see that we have x - (x + h); well, put -h in here...*2550

*Oh, I accidentally cut out the wrong thing there; I should not have crossed out the h; I should have crossed out the x.*2557

*The -x here cancels out the positive x here; and we are left with the limit as h goes to 0 of -h, divided by h times x times (x + h); great.*2567

*So, we have the h here on the bottom and the h here on the top.*2584

*So, we can cancel out h here and h here; that leaves us with -1 on the top:*2589

*the limit as h goes to 0 of -1, all divided by x times x + h.*2594

*At this point, if h goes to 0, we won't have massive issues.*2602

*Let's see one more: the limit as h goes to 0 of -1 over...we expand x and (x + h); we distribute that x over, and we get x ^{2} + xh.*2606

*Now, as h goes to 0, that will cause the xh to cancel out; but it will have no effect on the x ^{2},*2620

*so things don't end up breaking, as long as x isn't equal to 0.*2626

*But we are looking just in general; so -1/x ^{2} is what ends up coming out of this: -1/x^{2}.*2629

*So, that means that we can write, in general, f'(x) is equal to what we ended up getting of all of this in the end, -1/x ^{2}; great.*2636

*All right, finally, before we get to our final, fourth example, let's talk about the power rules.*2650

*These are the sorts of rules that you will end up learning as you start working through calculus.*2655

*One of the first and easiest rules to learn, that makes it so much easier to take a derivative, is the power rule.*2659

*What it says is that, for any function f(x) = x ^{n}, where n is just any constant number, the derivative of f(x) is f'(x) = n(x^{n - 1}).*2665

*So, we take that power; we bring it down in front of it; we have x to the n, and then we bring it down in front;*2680

*we have n times...and then we bring down our n to n - 1.*2690

*So, you might not have noticed this yet, but this actually ended up holding true*2695

*for all of the functions that we have worked through so far in this lesson, both in the early part of the lesson,*2699

*where we were setting up these ideas, and in Example 2 and Example 3.*2703

*When we took the derivative of f(x) = x ^{2} + 1, it came out to be simply 2x.*2707

*Don't be too worried about the +1; we can think of it is being x ^{0}.*2712

*So, when you bring down that 0 in front of it, it just cancels out and becomes nothing.*2716

*So, we got 2x out of it.*2719

*When we had f(x) = x ^{3}, when we took the derivative of that in Example 2,*2721

*we ended up getting 3 times x ^{2}: the 3 went down in front, and we brought down 3 by 1 to 2; 3 - 1 becomes 2.*2726

*And finally, Example 3, that we just worked on: f(x) = x ^{-1}...we brought down the -1, and we got -x; and -1 - 1 becomes -2.*2736

*So, we ended up seeing this inadvertently.*2746

*And this ends up being true for any constant number n; it really works for anything at all.*2749

*If you are curious about seeing this some more, try looking at what would happen if you tried to take the derivative*2755

*through that formal h goes to 0 of x ^{4}, f(x^{4}); try working through the formal definition of x^{4}.*2758

*And if you really want to try seeing how it works for any integer at all, or any positive integer, at least, try using the binomial theorem.*2765

*And it even works for anything at all, as we have just seen for negative numbers.*2773

*But you can end up seeing how it works for binomial theorem, as well.*2776

*If you are curious about this, try checking it out; it is pretty easy to end up seeing,*2779

*with the binomial theorem, that it ends up being true for any positive integer.*2782

*All right, let's put this to use: using that power rule, find an equation for the tangent line to the function f(x) = x ^{4} that passes through (3,81).*2785

*If we are going to pass through (3,81), let's first get a sense of what is going on.*2797

*Let's draw just a really quick sketch of what is going on here.*2800

*x ^{4} shoots up really, really quickly; we shoot up massively, very, very quickly, with x^{4}.*2803

*By the time we have made it to a horizontal x location of 3, we are putting out an output of 81; 3 ^{4} is 9 times 9, which is 81.*2810

*So, we are at this very, very high point, very, very quickly.*2818

*So, what we are going to want to find is: we want to find the tangent line to the function at this point, (3,81).*2822

*We want to find something that ends up going like this; that is what we are looking for: what is the slope at this?*2829

*Well, to find the slope at any given horizontal location, at any point, we end up taking the derivative.*2835

*How can we take the derivative? If we have f(x) = x ^{4}, the power rule says that we can get the derivative,*2842

*f'(x), by taking the exponent and bringing it down in the front; so we have 4 times x;*2852

*and the exponent goes...subtract by 1; so 4 - 1 becomes 4x ^{3}.*2861

*So, if we want to find out what the slope is at x = 3 (that is the horizontal location we are considering),*2867

*if we want to find out what the slope is for our tangent line, the slope at x = 3 is going to be f'(3).*2879

*f'(3) will be 4 times 3 ^{3}; we plug it into our f prime, our derivative function.*2885

*f'(3) = 4(3) ^{3}; 3 cubed is 27; 4 times 27 equals 108.*2893

*So, we now know that the slope of our tangent line is 108.*2902

*So, if we are going to work out the tangent line: our tangent line is going to end up using this slope, m = 108.*2907

*So, any line at all can be described by slope-intercept form; it is a really good form to have memorized: y = mx + b.*2917

*You always want to keep that one handy; it is always useful.*2925

*y = mx + b: well, we just figured out that the slope...f it is going to be the tangent, it must have a slope of 108, because f prime,*2928

*the instantaneous slope at 3, f'(3), comes out to be 108.*2938

*So, we know that the instantaneous slope at the point we are interested in, (3,81), is 108.*2942

*So, that means that we have y = 108x + some b that we haven't figured out yet.*2948

*How can we figure out what b is to finish creating our equation for the tangent line?*2957

*If you are going to figure out what any line is, you need to know*2962

*what the slope is and what the y-intercept is--what m is and what b is, at least if you are using slope-intercept form.*2966

*y = 108x + b; well, do we know any points on that line?*2972

*Yes, we are looking at the tangent line; and we were told that the tangent line passes through the point (3,81).*2977

*So, we can plug in the point (3,81), because we know that our tangent line*2984

*has to pass through the one point that it barely, barely just touches on that curve.*2989

*So, we plug in (3,81); that is 81 for our y-value; that equals 108 times 3 (for our horizontal value x) plus b.*2993

*81 = 108 times 3 (that comes out to be 324), plus b; 81 - 324 is -243 for our b.*3003

*So now, we know that -243 = b; we know what our slope is; so that means that we can describe the tangent line in general as y = 108x - 243.*3014

*That is y = mx + b with our m and b filled in.*3027

*And now, we have the tangent line that passes through the point (3,81) and is tangent to the curve created by x ^{4}--pretty cool.*3031

*Calculus gives us a whole bunch of stuff that we can end up doing with the derivative.*3039

*The derivative is this massively, massively useful, important thing.*3042

*And we are just barely touching the surface of how incredibly useful and cool this thing is.*3045

*As you work through calculus, you will end up learning a whole bunch of things that the derivative lets us do.*3050

*It lets us learn about a function; it is really, really amazing how much information it gives us; calculus is really, really cool.*3054

*I hope that, at some point, you get the chance to take calculus and get to see how many cool things there are.*3060

*And remember: when you work with the derivative, what you are looking at is what the slope is of that location,*3064

*of that point, of that horizontal location on your graph.*3068

*All right, we will see you at Educator.com later--goodbye!*3071

0 answers

Post by Kenosha Fox on March 16 at 01:16:57 PM

wow

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Post by James Whiddon on August 13, 2014

Excellent!

1 answer

Last reply by: Professor Selhorst-Jones

Fri Feb 14, 2014 11:17 AM

Post by DeAnna Dang on February 11, 2014

This helps me so much with my calc class!

1 answer

Last reply by: Professor Selhorst-Jones

Thu Jun 6, 2013 4:25 PM

Post by HAFSA Ahmad on June 6, 2013

wow !this session is excellent.