For more information, please see full course syllabus of AP Calculus AB

For more information, please see full course syllabus of AP Calculus AB

### Derivatives II

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
- Example I: Find the Derivative of (2+x)/(3-x)
- Derivatives II
- f(x) is Differentiable if f'(x) Exists
- Recall: For a Limit to Exist, Both Left Hand and Right Hand Limits Must Equal to Each Other
- Geometrically: Differentiability Means the Graph is Smooth
- Example II: Show Analytically that f(x) = |x| is Nor Differentiable at x=0
- Differentiability & Continuity
- How Can a Function Not be Differentiable at a Point?
- Higher Derivatives
- Example III: Find (dy)/(dx) & (d²y)/(dx²) for y = x³

- Intro 0:00
- Example I: Find the Derivative of (2+x)/(3-x) 0:18
- Derivatives II 9:02
- f(x) is Differentiable if f'(x) Exists
- Recall: For a Limit to Exist, Both Left Hand and Right Hand Limits Must Equal to Each Other
- Geometrically: Differentiability Means the Graph is Smooth
- Example II: Show Analytically that f(x) = |x| is Nor Differentiable at x=0 20:53
- Example II: For x > 0
- Example II: For x < 0
- Example II: What is f(0) and What is the lim |x| as x→0?
- Differentiability & Continuity 34:22
- Differentiability & Continuity
- How Can a Function Not be Differentiable at a Point? 39:38
- How Can a Function Not be Differentiable at a Point?
- Higher Derivatives 41:58
- Higher Derivatives
- Derivative Operator
- Example III: Find (dy)/(dx) & (d²y)/(dx²) for y = x³ 49:29

### AP Calculus AB Online Prep Course

### Transcription: Derivatives II

*Hello, welcome back to www.educator.com, welcome back to AP Calculus.*0000

*Today, we are going to continue our discussion of the derivative.*0004

*Let us jump right on in.*0007

*Let us start with an example, example 1.*0017

*I guess we will stick with black today.*0024

*It is not a problem, we will see how that goes. *0027

*Find the derivative of 2 + x/ 3 – x.*0030

*We know how to do this, nice straightforward.*0047

*f’(x) is equal to the limit as h approaches 0 of f(x) + h - f(x)/ h.*0050

*Again, we are going to work with just a function.*0062

*f(x) + h, it is going to be 2 + x + h/ 3 - x + h - 2 + x/ 3 – x.*0066

*That is that one and that one, /h.*0081

*It is like a little bit of a simplification to do.*0086

*This is just 2 + x + h/ 3 - x – h - 2 + x/ 3 - x/ h.*0088

*We are going to find a common denominator on top.*0105

*This is going to be 3 - x × 2 + x + h -, this × that,*0109

*-2 + x × 3 - x - h/ 3 - x - h/ 3 - x/ h.*0120

*I’m going to multiply it out.*0139

*I end up with something that looks like this, I hope.*0140

*+ 3x + 3h - 2x - x² - xh – 6 - 2x - 2h + 3x - x² - xh/ x.*0146

*I'm sorry, this is 3 - x - h × 3 – x, and all of that /h.*0180

*We get 6 - 6 because the negative distributes.*0191

*We get + 3x - the 3x, we get -2x.*0197

*A - and - 2x - x², - and - x² – xh.*0206

*What we are left with is just the 3h and the 2h.*0214

*3h - -2h, that is going to give us, it is going to equal 5h/ 3 - x - h × 3 - x/ h *0222

*which is going to equal 5h/ h × 3 - x - h × 3 – x.*0245

*The h cancel, you are just left with our function 5/ this thing which, now we are going to take the limit of that.*0258

*We get the limit as h goes to 0 of 5/ 3 - x - h × 3 – x.*0271

*h goes to 0 and we are left with 5/ 3 - x².*0288

*That is that, our f’(x) is equal to 5/3 – (x)².*0296

*Let us take a look at the graph and its derivative.*0310

*Again, the graph is in red, the derivative is in black.*0314

*As x increases, notice as x increases from left to right, *0320

*notice the slope of f, as it increases, the slope of the graph increases.*0341

*It is always positive which is why the black is positive above the x axis, but it is actually increasing which is why the black goes up.*0353

*3, of course.*0361

*What was our original f(x)?*0366

*Our original f(x) was 2 + x/ 3 – x, it is not defined at 3, there is an asymptote.*0369

*Our f’ was 5/ 3 - x², also not defined.*0378

*Here pass 3, the slope is always positive but notice now the slope is actually, *0386

*from your perspective, the slope is changing, it is decreasing.*0398

*Always going to stays positive but it is decreasing, getting close to 0.*0400

*That is why the derivative graph looks the way that it does.*0407

*The black graph.*0413

*As x increases from left to right, the slope of f, let me go back to black.*0417

*The slope of f is positive and increases at 3.*0428

*Neither f nor f’ exist, we said it is not differentiable there.*0445

*It is not defined there and it is not differentiable there.*0454

*The function is not defined, the derivative is not defined.*0458

*It is not differentiable.*0461

*We say f is not differentiable, it means something is going on there.*0465

*It is not smooth there.*0478

*It is not differentiable at x = 3.*0479

*Past 3, the slope is always positive above the x axis, that totally decreases.*0482

*Let us talk about differentiability.*0514

*We know the definition of the derivative.*0519

*If the limit that we take, when we take the derivative, the limit as h approaches 0 f(x) + h - f(x)/ h.*0522

*If the limit exists, the function is differentiable, in other words, it has a derivative.*0528

*If the limit does not exist, because we have seen situations where the limit does not exist, *0534

*we say function is not differentiable.*0538

*It does not have a derivative there at that point.*0539

*That is all that is going on.*0542

*We say that a function is differentiable at, a, if f’ at a exists.*0544

*In other words, f’ at a specific point is equal to limit as h approaches 0 of f of the point.*0580

*Now it is not just x + h, it is the actual point + h - the actual point/ h.*0593

*If f’ of a exists, in other words, if this limit actually exist as a finite number.*0610

*If this limit exists as a finite number.*0618

*Again, notice that we actually put a specific value in for this normal f(x) + h - f(x)/ h.*0638

*If we speak about differentially at certain point, we put the value in.*0646

*Actually, we did not have to do that.*0653

*The fact of the matter is we can just stick with the normal f(x) + h - f(x)/ h.*0655

*See if we actually get a derivative, some function of x, some f’(x).*0660

*And then, take the a and put it into the f’(x) function and see if it actually works out.*0665

*In the previous problem, we had f was 2 + x/ 3 – x.*0672

*And then, when we took the derivative of that, we ended up with f’ being equal to 5/ x - 3².*0680

*The question was, is it differentiable at 3?*0689

*It did not matter, you can go ahead and just use the normal general expression f(x) + h - f(x)/ h, take the limit.*0692

*When we took the limit, we ended up with an actual function.*0701

*Now we can put a in.*0704

*If you put 3 in here, you notice that 3 is not going to work, it is not defined.*0705

*Therefore, it is not differentiable.*0710

*It is up to you, you can put the a in, form the expression and take the limit.*0712

*Or you can just use the general expression as a function of x.*0716

*Find the limit if it exists, and then plug the a in and see if the derivative is actually defined, either one is fine.*0720

*I hope that made sense.*0728

*Let me just write this all out.*0730

*Notice we actually put the specific a value into the limit expression.*0732

*Or we can work more generally and say f(x) is differentiable if f’(x) exists.*0761

*That is it, the same thing, f’(x) = the limit as h approaches 0 of f(x) + h - f(x)/ h.*0813

*If you take the limit and you get a function.*0826

*It is going to exist for some values of a function, for certain values.*0831

*It is not going to exist so it is not going to be differentiable at those particular points, *0834

*where the derivative f’(x) is not defined or if there are some other problem there.*0839

*Recall, for a limit to exist both left hand and right hand limits have to be the same.*0860

*Right hand limits must equal each other.*0883

*Graphically it means this.*0893

*That is our a, we have a secant line.*0909

*This point is a + h.*0920

*The limit as h goes to 0 of f(a) + h – f(a)/ h.*0926

*That is the slope of that line.*0933

*As I get closer and closer, I’m going to get, this is the secant line.*0936

*I’m going to have another secant line, another secant line.*0944

*At some point, it is going to get so close, I’m going to end up with a tangent line.*0953

*The same thing from this side, if I have a secant line and if I get closer and closer,*0956

*I’m going to have a bunch of other secant lines that pass through that point.*0963

*You see the secant line slope is approaching a certain value.*0972

*In the other end, the secant line slope is approaching a certain value.*0977

*If the two lines match the left hand and the right hand, that is what it means for the slope to exist.*0981

*The left hand slope and the right hand slope have to equal.*0991

*The left hand limit and the right hand limit have to equal.*0994

*If they equal each other, we end up with the derivative.*0997

*Now if I plug in a value of a and calculate it, the two numbers have to equal each other.*1000

*It has to exist first of all, but they have to equal each other.*1008

*If I do it with general function, find the function and that function is going to be undefined somewhere.*1012

*I can do it either way.*1019

*I can either use the expression with a specific value of a or I can use just the general expression with x and put a in afterward.*1021

*But this is what it means for a limit to exist.*1030

*The slopes have to match from left and right.*1033

*Geometrically, this means, as you pass to the limit as h approaches 0, *1036

*the secant line slopes from the left and the right, they approach the same numerical value.*1062

*In other words, the secant lines become the same tangent line.*1094

*Same numerical value means the same tangent line.*1102

*Geometrically, this means, the graph is smooth.*1114

*It does not have any kinks in it.*1118

*Geometrically, this idea of differentiability means the graph is smooth.*1122

*Smooth means no sudden jerks, no sudden changes of direction, or going off to infinity.*1149

*When dealing with graphs, a differential function, any point that all of the sudden has a sudden of change of direction,*1176

*the function is not differentiable at that point.*1186

*Any point x where the graph goes off to infinity, the graph is not differentiable.*1189

*Differentiable means smooth, nice transitions.*1194

*It goes off to infinity.*1198

*You have something like this.*1200

*Notice, all, everything, there is no sudden change of direction.*1207

*If you have something like this, it is not differentiable there and it is not differentiable there.*1210

*The slope all of a sudden goes from here to here.*1220

*The slope was from here to here.*1225

*No cusp, any cusp on the graph means not differentiable.*1232

*If you have a situation where it goes to infinity, it is not differentiable at that x value.*1238

*Example 2, this one is a bit involved, but that is not a problem.*1253

*It is a really great example.*1261

*Example number 2, show analytically that f(x), the function f(x) = the absolute value of x, is not differentiable at x = 0.*1262

*We know what this graph looks like.*1291

*Here is our graph, it is just going to be that thing right there.*1294

*It is perfectly differentiable here, perfectly smooth and perfectly smooth.*1297

*Notice here it has a sudden change of direction.*1301

*The slope and the slope are not equal.*1304

*Geometrically, because we can see that it is not differentiable.*1306

*We want to show analytically that it is not the case.*1309

*We have to show that the left hand limit and right hand limit, even though they exist, they are not equal to each other.*1313

*Remember, in order for there to be differentiability, the limit has to exist.*1324

*For the limit to exist, the left hand and the right hand limits have to be equal to each other.*1330

*I hope that make sense.*1338

*We must show that f’ at 0 does not exist, that is non differentiability.*1340

*In other words, we have to show that it is infinite or the left hand limit does not equal the right hand limit.*1356

*We have our f(x), f(x) = the absolute value of x.*1373

*0 is the dividing line, they also told us that we are concerned with 0.*1378

*We must check x greater than 0 and x less than 0, and compare the limits.*1388

*In other words, we are going to do the limit as x approaches 0 from below.*1406

*We are going to take the limit as x approaches 0 from above.*1412

*We are to see what those two limits are.*1416

*If the limits are the same, it is differentiable.*1418

*If they are not the same, it is not differentiable.*1421

*This is how we do it analytically.*1423

*We have to show the limit exists, we have to show that limits equal each other.*1425

*For x greater than 0, we have f’(x) is equal to the limit as h approaches 0 from above of f(x) + h - f(x)/ h =*1433

*the limit as h approaches 0 from above of the absolute value of x + h - the absolute value of x/ h.*1469

*The absolute value of x + h, when x is greater than 0, recall the definition of absolute value.*1482

*Let me do it another way, the absolute value of anything under the absolute value sign is that thing, *1492

*when x is greater than 0, when a is greater than 0.*1497

*Or it is –a, when a is less than 0.*1500

*Since x is greater than 0, this just becomes the limit as h approaches 0.*1503

*The absolute value signs go away.*1510

*It is just x + h - x/ h.*1512

*x cancel and you are left with the limit as h approaches 0 from above of 1 which = 1.*1517

*I hope that makes sense.*1529

*For x less than 0, f’(x) is equal to the limit as h approaches 0 from below of f(x) + h - f(x)/ h = *1535

*the limit as h approaches 0 from below.*1566

*This is x is less than 0.*1571

*This thing is actually going to be –x + h - -x.*1579

*Because the absolute value of x, when x is less than 0 is –x, that minus stays.*1591

*I hope that makes sense.*1599

*/h, = the limit as h approaches 0 from below - x - h + x/ h.*1605

*= the limit as h approaches 0 from below -1.*1619

*The left hand limit, when we approach 0 is -1.*1627

*The limit exist, it is -1.*1631

*The right hand limit as x approaches 0 from above was +1.*1633

*The left hand limit exists, the right hand limit exists,*1640

*but the limits are not equal to each other which means that it is not differentiable.*1642

*-1 does not equal 1.*1649

*f(x) is not differentiable but it is differentiable everywhere else.*1655

*It is differentiable from negative infinity all the way to 0, union 0 all the way to positive infinity.*1666

*It is differentiable everywhere else, because everywhere else is defined.*1675

*As long as x is less than 0, the slope is always going to be -1.*1678

*As long as x is greater than 0, the slope is always going to be +1.*1684

*0, it cannot decide which slope to take, -1 or +1.*1689

*Because it cannot decide, because they are not equal, it is not differentiable.*1693

*It is not smooth, there is a sudden change of direction.*1697

*It is not differentiable, that point causes problems.*1704

*That is all that is going on here.*1708

*Let us go ahead and draw these out.*1719

*We know what f looks like, this is f(x) = the absolute value of x.*1725

*It is not differentiable here, it is not smooth there.*1732

*The graph of the derivative f’(x).*1736

*When it is bigger than 0, we said the slope is 1.*1743

*When it is less than 0, the slope is -1.*1748

*Notice it is not defined here.*1750

*The f’(x) is not defined here.*1753

*This is a discontinuity, it is not differentiable, it is not smooth.*1757

*The function is continuous but the function is not differentiable.*1763

*It is okay, you can have a function that is continuous.*1769

*Notice change direction, I do not have to lift my pencil off.*1771

*This is not the f graph, this is the f’ graph.*1776

*Be very careful, this is going to be probably the single biggest problem for a couple of weeks *1781

*until you just learn to separate the fact that you can have a function and *1787

*you can have the derivative of the function also be a function.*1792

*You are going to be graphing both, keeping the graphs separate.*1795

*Now in calculus, my best advice is slowdown, be very careful.*1799

*There is going to be a lot of stuff going on in the page.*1803

*We will have symbolism.*1806

*This symbolism is going to be very subtle.*1807

*The difference between f and f’.*1809

*If your eye tends to move quickly, it is going to miss it.*1811

*The function itself, the original function is continuous.*1814

*I do not have to lift my pencil up.*1817

*The graph of the function shows that it is not differentiable.*1819

*It is not smooth there.*1824

*It does not go like this.*1826

*Let us formalize that notion.*1845

*Let me draw the graph again.*1846

*This us our coordinate axis, this is our f(x) = the absolute value of x.*1850

*What is f(0), f(0) is just 0.*1861

*The absolute value of 0 is 0.*1870

*What is the limit as x approaches 0 of the absolute value of x?*1873

*It is also 0.*1884

*Let me break this one up actually.*1892

*The limit is x approaches 0, we have to do the left and right hand limits.*1897

*The limit as x approaches 0 from above of the absolute value of x = the limit as x approaches 0 from above of x.*1901

*Because when x is positive, x approaches 0 from the right, all positive numbers.*1910

*The absolute value of x is positive = 0.*1916

*The limit as x approaches 0 from below of absolute value of x = the limit as x approaches 0 of –x.*1921

*When we are dealing with negative numbers, it is approaching 0 there.*1931

*The absolute values of x is –x.*1937

*Here I still get 0.*1938

*All three are equal, the left hand limit, the right hand limit, and the value of the function.*1943

*Therefore, the function is continuous.*1950

*What I just demonstrated is the analytical way of showing that the absolute value function is continuous.*1952

*All three are equal.*1960

*f(x), the absolute value function is continuous at x = 0.*1966

*Notice what I have done, what I have done here, the limit of the function as x approaches 0, that is not what I just did.*1975

*What I just did is I found the limit as h approaches 0 of f(x) + h - f(x)/ h.*1985

*I took the derivative.*2001

*The derivative means finding the limit of this quotient.*2003

*Here, as h approaches 0, here I’m taking the limit as x approaches 0 of the actual function itself.*2008

*When I’m taking the limit of the function itself, I’m checking for continuity.*2015

*When I’m finding the limit of this thing called the Newton quotient, I'm finding the derivative.*2019

*We see that analytically, we have demonstrated that the function is continuous.*2024

*And previously, we demonstrated analytically but is not differentiable.*2029

*You can have a function, let me say that again.*2042

*I think it is very important to say that again.*2056

*The limits that we just evaluated, those were the limits as x approaches 0 of f(x).*2061

*The limits that we evaluated prior to that to show differentiability or non differentiability, *2069

*those were the limit as h approaches 0 of f(x) + h - f(x)/ h.*2076

*This concerns continuity.*2086

*This concerns differentiability.*2089

*Do not mistake the two.*2091

*I hope the last two things did not confuse you.*2094

*You can have a function be continuous at a point a, but not differentiable at that point at a, as we just saw.*2101

*However, if f(x) is differentiable at point a, then it is continuous at a.*2137

*Differentiability implies continuity.*2164

*But continuity does not imply differentiability.*2168

*If I know a function is differentiable at 5, I know that it is continuous at 5.*2172

*I’m given continuity automatically, as a free gift.*2175

*But if I'm only continuous at 5, that tells me nothing about whether it is differentiable at 5.*2179

*Continuity does not give me differentiability for free.*2185

*Different ability gives me continuity for free.*2188

*Because sometimes, we are going to know some function is continuous, we are not going to know it is differentiable.*2191

*Other times, we are going to know a function is differentiable, we automatically going to know that it is continuous.*2195

*Differentiability implies, remember this double headed arrow means implies continuity.*2202

*In other words, if it is differentiable then it is continuous but not the other way around.*2217

*Notice it is not a double headed arrow.*2223

*If it were a double headed arrow, it would mean it works this way and it works that way.*2226

*Let us see what we have got, what is next.*2247

*Differentiability implies continuity but not the other way around.*2253

*In case this idea of implication, differentiability implies continuity but it does not go the other way.*2260

*If it did, we have a double headed arrow.*2271

*Now let us get some real world examples of what this idea of implication means.*2274

*When we say if something is true then something else is true.*2278

*If I have, if a then b, that does not mean if b then a.*2281

*It means only if a then b.*2286

*If it were true the other way around, let us call it converse.*2289

*I would specify if b then a.*2292

*They are definitely different.*2296

*Let me give you a couple of real world examples, in order to help with this intuitive notion of what implication actually means.*2297

*Real world example of this thing which is the logical implication.*2305

*It is huge in mathematics because everything is about if then.*2319

*Now I can write, if rain then clouds.*2322

*In other words, rain implies clouds.*2329

*What that means is, if it is raining, I know automatically it is cloudy.*2331

*There is no other way it is going to happen.*2337

*However, just because it is cloudy, it does not mean that is raining, that is what this means.*2338

*It works one way, it does not work the other way.*2344

*We are very specific in mathematics about this.*2346

*Clouds do not imply rain, but rain always implies clouds.*2351

*Differentiability implies continuity.*2365

*In other words, if it is differentiable, we automatically know it is continuous.*2366

*But just because something is continuous, it does not mean that it is differentiable.*2370

*How can a function fail to be differentiable?*2377

*What we just saw.*2379

*How can a function fail to be differentiable?*2382

*How can a function not be differentiable at a point?*2388

*The first way is a sharp kink.*2403

*I should say a sharp change of direction.*2406

*Sharp change of direction, that is one way.*2411

*If we have a sharp change of direction like the absolute value function, it is not differentiable there.*2415

*A discontinuity in the original function of the derivative, the f(x).*2419

*Three, an infinite slope.*2438

*If you have an infinite slope, in other words, the slope is all of a sudden goes vertical.*2445

*Change in y/ change x, change in 0, it is infinite.*2452

*Vertical line, whatever vertical line, there is no slope.*2455

*The slope is undefined, it is not differentiable there.*2457

*If you have something like rapid change in direction, it is not differentiable at that point.*2463

*It is not differentiable at whatever x value is.*2471

*A discontinuity function is not differentiable at that point.*2475

*The last one, if you have a function that goes vertical and it was like that, *2486

*the slope goes vertical, it is not differentiable at that point.*2492

*Three ways you can fail differentiability.*2497

*Sharp change in direction, a discontinuity, or an infinite slope.*2499

*Let us talk about the final topic here.*2506

*Let us see where we are, maybe higher derivatives.*2509

*I think I will go to blue, just a happy color and makes me happy.*2516

*Higher derivatives.*2522

*I know that if I have a function f(x), I can take the derivative of it and I get some f’(x) which is also function of x.*2530

*I can take the derivative of that.*2537

*As a function of x, I can take the derivative again.*2539

*I can take the derivative as many times as I want.*2542

*As long as I end up with something that is actually meaningful.*2545

*I get the second derivative which we symbolize with a double prime and a triple prime,*2548

*and a quadruple prime, however many derivatives you can actually take.*2553

*y, take the derivative, you get y’.*2558

*Let us do it this way. If I have a function at y, if I take the derivative, dy dx, if we are using that symbolism.*2563

*Now when we take the second derivative, when we take the derivative again,*2570

*the symbol is d² y dx².*2575

*If it was the 4th derivative, it would be the d⁴ y/ dy⁴.*2580

*This is a symbol telling me that I have taken the derivative twice.*2586

*Let us talk about this.*2591

*Let us talk about this peculiar symbolism.*2599

*What the heck is d² y/ dx² mean, why do we symbolize it that way.*2606

*If you do not want to do this, yes I do.*2617

*In mathematics, we have something called an operator.*2623

*An operator is a symbol that tells you do something to a function.*2632

*A symbol that tells you to perform an operation on the symbol.*2641

*It is a fancy word for, it is a symbol to tell you to do something to a function, to perform an operation on a function.*2654

*Given f, taking the derivative of f is an operation that you are performing on the function f.*2664

*It is an operation you perform on f, in order to get f’.*2690

*You have a function, you operate on that function with the differential operator, *2705

*with the derivative operator, and you get another function.*2708

*The derivative operator is symbolized ddx.*2715

*Whenever you see d/ dx and there is something that follows it, that means take the derivative of what is following it.*2732

*When you see ddx of x², this says take the derivative of x².*2744

*It is that simple, it is just take the derivative of x².*2771

*Now we said y is also a symbol used for functions.*2775

*We have seen it for years now, ever since probably 6th or 7th grade.*2793

*y = x², we see that all the time.*2806

*It can be f(x) = x² or we just write y = x².*2809

*Dy dx is equivalent to ddx of y.*2817

*We just decide to get rid of the parentheses and put the y there, up on top with the numerator.*2825

*Dy dx which is equivalent to ddx of y, which is equal to ddx of the function x² says, take the derivative of y.*2831

*That is it, take the derivative of y.*2845

*When you see dv dt, it is telling you z is some function of some variable.*2850

*Take the derivative with respect to t of that.*2861

*It is telling you that z is actually some function of t.*2865

*It is telling you take the derivative of z.*2869

*Dy dx, take the derivative of y.*2872

*That is all it is, find f’, find y’, whatever it is.*2875

*Given y, apply the differential operator that gives you dy dx.*2882

*Apply the differential operator again, in other words take the derivative again, *2897

*that gives you, you are taking ddx of this thing which is a function.*2902

*You are doing ddx of dy dx, multiplied out symbolically.*2908

*d × d is d², dx dx we just put the 2 on top of that.*2918

*d² dx, this is actually the whole thing dx.*2926

*I hope that makes sense.*2932

*That is where the symbolism comes from.*2933

*It is just the idea of applying this thing called an operator.*2937

*In this case, it is the derivative operator.*2941

*Later in the course, you are going to learn something called the integral operator*2943

*which means take the integral of the function, instead of take the derivative of the function.*2946

*Symbolically, that is all it means.*2955

*It is all based on this symbol, an operator which says do something to a function.*2956

*When you have done something to a function, you are going to get another function.*2962

*Let us do example 3, find dy dx and d² y dx² for y = x³.*2970

*The first derivative, we need to apply the differential operator once to get something.*2993

*And then, we need to apply the differential operator again to get something else.*3002

*Again, applying the differential operator just means taking the derivative.*3005

*Finding the limit as h approaches 0 of f(x) + h - f(x)/ h².*3009

*In this case, when we do it, this is our f(x).*3016

*Once we get the first derivative, now we are going to treat this as a function of x and*3021

*we are going to ignore this and treat this as an f(x).*3026

*We are going to be taking the limit of f of this, the derivative.*3029

*The first derivative, the first derivative is just f’(x) is equal to the limit as h approaches 0 of f(x) + h - f(x)/ h.*3037

*We did this already in the previous problem.*3057

*I do not want to go through this process again.*3058

*Let us just say because we are dealing with y = x³, we found that f’(x) is equal to 3x².*3061

*That is our first derivative.*3070

*Now we want to take the second derivative.*3072

*The second derivative means take the derivative now of 3x².*3075

*f”(x) is equal to the limit as h approaches 0 of f’(x) + h - f’(x)/ h.*3084

*This is f, the derivative give me a prime.*3102

*Here is f’, this is f’’.*3106

*It is a prime on top of that prime, that is what is happening here.*3113

*This is going to be the limit as h approaches 0 of this function 3x + h² - 3x²/ h.*3117

*When I expand this and simplify it, I end up with 6x.*3136

*You should do it for yourself to actually make sure that it works.*3144

*What you have got is x³.*3150

*I can go to the next page.*3159

*x³, the first derivative.*3162

*When we take the derivative of that, we get 3x².*3168

*We take the derivative again and we get 6x.*3172

*If I wanted to, I can take the derivative again.*3176

*I would get 6.*3179

*If I wanted to, I can take the derivative one more time and I would end up with 0.*3180

*In this case, this is differentiable four times.*3184

*You will actually see how we go from this to this to this to this.*3188

*For the time being, we have been using the limit as finding the derivative.*3193

*Our process of finding the derivative so far is taking the limit as h approaches 0 of f(x) + h.*3200

*We actually go through this tedious algebraic process.*3208

*In the subsequent lessons, we are going to teach you quick ways to go from here to here,*3212

*without having to go through this process.*3218

*Thank you so much for joining us here at www.educator.com.*3222

*We will see you next time, bye.*3224

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