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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 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

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 operator2943

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 and3021

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