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

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

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## Transcription

### More Chain Rule Example Problems

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: Differentiate f(x) = sin(cos(tanx))
- Example II: Find an Equation for the Line Tangent to the Given Curve at the Given Point
- Example III: F(x) = f(g(x)), Find F' (6)
- Example IV: Differentiate & Graph both the Function & the Derivative in the Same Window
- Example V: Differentiate f(x) = ( (x-8)/(x+3) )⁴
- Example VI: Differentiate f(x) = sec²(12x)
- Example VII: Differentiate
- Example VIII: Differentiate
- Example IX: Find an Expression for the Rate of Change of the Volume of the Balloon with Respect to Time

- Intro 0:00
- Example I: Differentiate f(x) = sin(cos(tanx)) 0:38
- Example II: Find an Equation for the Line Tangent to the Given Curve at the Given Point 2:25
- Example III: F(x) = f(g(x)), Find F' (6) 4:22
- Example IV: Differentiate & Graph both the Function & the Derivative in the Same Window 5:35
- Example V: Differentiate f(x) = ( (x-8)/(x+3) )⁴ 10:18
- Example VI: Differentiate f(x) = sec²(12x) 12:28
- Example VII: Differentiate 14:41
- Example VIII: Differentiate 19:25
- Example IX: Find an Expression for the Rate of Change of the Volume of the Balloon with Respect to Time 21:13

### AP Calculus AB Online Prep Course

### Transcription: More Chain Rule Example Problems

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

*Today, I thought we would do some more chain rule example problems.*0005

*The chain rule was one of those things that you are pretty much going to be using all the time in your differentiation.*0008

*You are never going to be dealing with straight simple functions.*0015

*It can be a little bit deceiving, in the sense that you think you have a grasp on it.*0019

*And yet somehow things tend to slip away because if you forget to take that last derivative or whatever it is.*0025

*In any case, I just thought it would be nice to do some more example problems.*0030

*Let us jump right on in.*0033

*Nothing particularly difficult, just more practice, something nice to see.*0036

*Differentiate f(x) = sin of cos(tan(x)).*0041

*We have the sin, the cos, the tan.*0049

*We have 3 things, we are going to end up with 3 terms in our derivatives.*0051

*That is how it works.*0054

*The number of things that you actually have, the number of actual functions,*0055

*that is how many terms you end up with in your derivative.*0058

*A good way to think about it if you lose your way.*0061

*I will stick with black or blue, we will stick with blue.*0066

*F’(x) = we are going to take the sin, the derivative of the sin.*0071

*We are working outside in.*0075

*It is going to be the cos of what is inside.*0078

*What is inside is the cos(tan(x)).*0082

*Now we take the derivative of what is inside.*0087

*The derivative of the cos(tan(x)) is –sin tan(x).*0091

*Now we take the derivative of the tan(x) which is sec² x.*0098

*That is it, we are done.*0104

*F’(x), if you want to go and bring this negative sign out front, you are going to end up with f’(x) = -cos of the cos(tan(x)) × sin(tan(x)) × sec² x.*0106

*Is there a way to simplify that, maybe or maybe not?*0128

*That it is not really worth it.*0131

*You are just going to go ahead and use the function as is.*0134

*Nothing particularly strange, you just have to follow through and make sure you go down the line, go down the chain.*0137

*Example number 2, find an equation for the line tangent to the given curve at the given point.*0145

*Our function is sin(x)² π/2.62438.*0153

*Let us go ahead and find the derivative.*0159

*F’(x) is equal to, the derivative of sin is cos.*0162

*It is going to be cos(x)² × the derivative of what is inside, 2x.*0169

*Therefore, if I want I can that 2, it does not really matter.*0178

*Now we are going to go ahead and put π/2 into the x, to actually go ahead and find what this slope is.*0184

*The f’(π/2) = cos(π/2)² × 2 × π/2.*0194

*You end up with, these two cancel here.*0210

*You end up with cos.*0214

*Some number I went ahead and converted it to degrees, just for the heck of it.*0216

*141.7°, it is just going to be π²/ 4.*0221

*The cos(π²)/ 4 is just going to give you some number, and then × π.*0225

*When we solve that, we end up with -0.78 π.*0230

*This is our slope, we want the equation of the line, that is just y - y1 which is 0.6243 = m,*0237

*the slope -0.785 × x - x1, which is –π/2.*0248

*There you go, you can leave it like that, not a problem.*0256

*Example number 3, F(x) is equal to f(g(x)).*0264

*They give a certain values f(4), f(6), f’(4), g(6), g’(6), find f’(6).*0269

*F(x) is f(g(x)), therefore, f’(x) by the chain rule is f’ at g(x) × g’(x).*0277

*Therefore, f’ at 6 = f’ at g(6) × g’(6).*0293

*This equals, g(6) is equal to 4.*0305

*This is going to be f’(4) × g’(6).*0312

*This is f’(4) is equal to 2 and g’(6) is equal to 9.*0318

*Our final answer is 18.*0327

*That is it, just working it out, plugging it in.*0330

*Nice and straightforward.*0333

*Differentiate this function and then graph both the function of the derivative in the same window.*0337

*I just thought that I will throw that extra graphing thing in there.*0343

*Let us see what we have got.*0348

*We have f(x) = this.*0350

*It looks like we are going to be doing the quotient rule.*0353

*This × the derivative of that - that × the derivative of this divided by this².*0358

*F’(x) is going to equal this × the derivative of that.*0367

*That is going to be x × ½ × 1 – x³⁻¹/2 × the derivative of this thing, we still have to take the derivative of that.*0372

*That is going to be × -3x², this × the derivative of that.*0387

*Yes, very good, - that × the derivative of that.*0404

*The derivative of this is 1.*0409

*It just stays 1 – x³¹/2.*0412

*And then, × the derivative of that which is 1.*0419

*We will take all of that /x².*0422

*I’m going to go ahead and rewrite this.*0429

*I will do -3x².*0433

*I’m going to put this and I'm going to go ahead and bring this down to the bottom.*0438

*I’m going to put these, it is going to be -3x³ 3x² x divided by, I will leave the 2.*0442

*I’m going to bring this down below.*0451

*I’m going to actually rewrite it as a radical.*0453

*This is going to be √1 – x³.*0456

*This is going to be –√1 – x³.*0460

*All of that is going to be /x².*0470

*I get myself a common denominator here.*0475

*The common denominator is going to be 2 × 1- √x³.*0482

*We get -3x³ – 2 × 1 – x³.*0488

*Because when I multiply this × that, the radical sign goes away.*0498

*All over 2 × √1 – x³/ x².*0505

*Here we are going to have – 3x³.*0512

*A – and – x³, this is going to be -2x³ – 2/ 2 × √1-x³/ x².*0518

*I will just go ahead and bring this x² and put it in the denominator.*0542

*I’m left with -2x³ – 2.*0544

*If I did my arithmetic correctly, which is always the task up.*0548

*2x², 1 – x³.*0552

*There you go, this is our derivative.*0559

*If I did the algebra correct.*0561

*As far as what it looks like, it basically just looks like this.*0564

*This red one, this is the original f(x).*0570

*This little light blue one, this is f’(x).*0574

*F(x), this is f(x), f’(x) and f’(x), there you go.*0579

*This way, you see the slope is positive.*0587

*If it is positive, it is above the x axis.*0591

*At some point, it actually hits 0, becomes horizontal.*0594

*And then, the slope becomes negative, that is the derivative of the line.*0596

*Here the slope is negative, starts to become a little positive.*0599

*It zeros out again and it actually drops back down.*0605

*It starts to get a little positive but then it drops back down.*0608

*That is it, straight looking graph but these are the functions that we are going to be dealing with in the real world.*0611

*Differentiate f(x) = x – 8/ x + 3⁴.*0621

*Here, f’(x), chain rule.*0628

*We will do 4 × x – 8/ x + 3³ ×, the derivative of what is inside.*0633

*The derivative of what is inside is going to be quotient rule.*0645

*It is going to be x + 3 × the derivative of the top which is 1, -x – 8 × the derivative of the bottom which is 1, all over x + 3².*0649

*This is going to equal, let me bring it down here.*0666

*This is going to equal 4 ×, I‘m going to separate this one out.*0672

*It is going to be 4 × x³/ x + 3³ ×,*0680

*Here we have x + 3 – x – 8.*0690

*I will just write it out, it is not a problem.*0699

*It is going to be x + 3 – x + 8.*0701

*Sorry about that, it is going to be –x.*0706

*A – and -8 is going to be +8.*0708

*This is going to be x + 3, it is going to be².*0712

*This is going to equal, the x – x, they go away.*0716

*We have 3 and we have 8.*0720

*3 + 8 is 11, 11 × 4 is 44.*0726

*On the top we have 44 and we have x – 8³.*0731

*The bottom x + 3³, x + 3².*0735

*We get x + 3⁵, there we go.*0738

*Just have to follow the chain.*0746

*F(x) = sec² 12x.*0751

*This one is really nice and simple.*0754

*Let us see, sec² 12x.*0761

*We have to deal with this one first.*0766

*This is the same as, the sec(12x)².*0768

*We have to do chain rule first.*0776

*It is going to be 2 sec 12x.*0779

*Now we take the derivative of the sec which is going to be sec 12x tan 12x.*0793

*And then, × the derivative of the 12x, there is that.*0805

*Our final answer is going to be, I will take the 2 and the 12 together, it is going to be 24.*0809

*Sec 12x, I will put those together.*0815

*Sec² 12x tan 12x, that is our final answer.*0819

*We have three things going on.*0831

*We have the power function, we have the sec function, and we have the 12x function.*0832

*There are 3 three things so we are going to have 3 terms in our derivative.*0838

*Here is the first term, that takes care of the second term.*0842

*This takes care of the third term.*0850

*Just keep it that way.*0852

*If you want, you can draw the lines in between to separate your terms.*0855

*Identify how many actual functions you have, and then that the terms.*0859

*If you have counted 1, 2, 3, and then you only ended up with two, you know that you are going to need one more.*0864

*That one more is going to be the derivative of that.*0869

*As many functions as you have in your main function, that is how many terms*0871

*you are going to have in your derivative of the function.*0876

*Differentiate this, e ⁺x² + 3x × sin(x²) + 3x/ cos x.*0884

*This is going to be long and painful.*0894

*Maybe not too bad, let us see what is going on here.*0899

*Let us go back to blue.*0902

*We have f’(x), probably going to run out of room but it is not a problem.*0904

*It is going to be this × the derivative of that – that × the derivative of this/ that².*0910

*This × the derivative of this.*0916

*We have cos x × the derivative of this.*0919

*The derivative of this, the top is going to be product rule.*0924

*It is going to be this × the derivative of that + that × the derivative of this.*0927

*Here is what we end up getting.*0934

*It is going to be e ⁺x² + 3x × the derivative of that.*0942

*The derivative of that is going to be cos(x²) + 3x × the derivative of what is inside*0949

*which is going to be 2x + 3 + that × the derivative of this.*0959

*It is going to be + sin(x²) + 3 × the derivative of this.*0966

*The derivative of that is equal to e ⁺x² + 3x ×, I will just bring it down here, the derivative of that which is 2x + 3.*0977

*That is the first part, this × the derivative of that.*0997

*From that, this is still the numerator here, -this × the derivative of this.*1001

*It is going to be –e ⁺x² + 3x × sin(x²) + 3x × the derivative of this × -sin x.*1012

*It is going to be all of this /that².*1028

*What are we going to do with this?*1039

*Probably, you do not want to simplify it too much.*1040

*I’m going to go ahead and pull out a, 2x + 3, I have a 2x + 3, I’m going to pull that out.*1043

*I have an e ⁺x² + 3x, that is one term.*1055

*I have an e ⁺x² + 3x, that is another term.*1058

*I have an e ⁺x² + 3x, that is another term.*1063

*Let us see, let me just take this first part here.*1067

*This first part, let me deal with that.*1076

*I’m going to go ahead and factor out a 2x + 3.*1079

*I’m going to factor out an e ⁺x² + 3x.*1085

*I’m also going to factor out a cos(x).*1092

*That is going to be cos(x)² + 3x + sin(x)² + 3x.*1096

*I’m going to deal with this term right here.*1116

*I’m just going to actually leave it alone, - and – becomes +.*1118

*It becomes + e ⁺x² + 3x × sin(x)² + 3x × sin(x).*1121

*All of that is the numerator, and then, cos² x.*1138

*I think it is best to just leave it at that point, just one little step of simplification.*1144

*You could have left it at that, it is not a big deal.*1150

*Just make it a little bit cleaner, if nothing else.*1152

*There you go, very complicated as we would expect.*1156

*We have a very complicated looking function.*1159

*Exponential, trigonometric, there is going to be a lot going on.*1162

*Differentiate f(x) = 6 ⁺sin(π/2x).*1167

*Let us recall, anytime we have a constant raised to some power,*1175

*the derivative of that = a ⁺u and then the nat-log of the base which is a.*1182

*F’(x) that is going to equal 6 ⁺sin(π/2x) × ln (6) × du dx, chain rule.*1189

*Now we take the derivative of this.*1209

*The derivative of sin(π/2x) is cos(π/2x) × the derivative of π/2x which is π/2.*1213

*We can put some things together here.*1226

*Let me go back to blue.*1230

*Ln (6) is a constant, π/2 is a constant.*1232

*Let us write it as π ln 6/ 2 × this thing × this thing ×,*1236

*Let me go ahead and put a parenthesis.*1253

*6 ⁺sin(π/2x) × cos(π/2x).*1255

*There we go, that is our final answer.*1266

*This one right here, the rate of change of the radius of a spherical balloon with respect to time as we inflate it is 1.5 cm/s,*1274

*find an expression for the rate of change of the volume of the balloon with respect to time.*1285

*The rate of change of the radius of the spherical balloon with respect to time, that is dr dt, that is equal to 1.5 cm/s.*1290

*Basically, you have this balloon, you are inflating it, it is getting bigger and bigger.*1306

*This radius is changing, for every second it is growing by 1.5 cm.*1313

*That is all this says.*1320

*It is the cm/s, radius is in centimeters, time is in second.*1321

*This is a rate of change, that is how we express the rate of change.*1326

*Dy dx is the rate of change of y for every unit change in x.*1329

*For every unit interval of time, 1 second, the radius changes by 1.5 cm.*1333

*That is all this is, it is just the rate.*1341

*They want to know, find an expression for the rate of change of the volume with respect to time.*1344

*What they want is dv dt.*1348

*We have a relationship between v and r.*1352

*Let us go ahead and deal with that.*1355

*We know that the volume of a sphere is 4/3 π r³.*1358

*We have dr dt, we want dv dt.*1369

*Dv dt, we will just differentiate everything with respect to t.*1370

*Dv dt = 4/3 π ×, r is a function of t.*1376

*We treat it chain rule, it is going to be × 3 r² dr dt.*1386

*The 3 and the 3 cancel, therefore, I get dv dt = 4 π r² dr dt.*1395

*Dr dt is 1.5, I plug that in here.*1410

*I get 4 π r² × 1.5.*1415

*The final answer is, the rate of change of volume with respect to time is equal to 6 π r².*1422

*Whatever the radius happens to be at moment, I put that into this equation.*1435

*That will tell me, at that moment, how fast the volume is changing.*1439

*Notice, here, the rate of change of the radius is constant.*1443

*That means the radius is growing by 1.5 cm every second, it is not changing.*1448

*But every second it is 1.5 cm, another second 1.5 cm, another second 1.5 cm.*1453

*The volume is not linear, it is not constant.*1458

*It actually depends on what r is.*1462

*As r increases, the rate at which the volume is changing actually keeps getting more and more.*1465

*Because now, the rate of change of volume depends on r.*1474

*As r changes, the square of r, the number is going to change.*1480

*If r is 4, this is going to be 4 × 4 is 16, 6 × 16 π.*1483

*When the radius is 4, the volume is changing at 6 × 16 π.*1488

*If the radius is 10, 10² is 100, 6 × 100.*1494

*When radius is 100, the volume is changing per second at 600 π.*1500

*The rate of change of the radius is constant but the rate of change of volume is not constant.*1508

*It actually depends on r.*1515

*This is the general procedure.*1518

*You will be seeing a lot of this when we do related rates.*1519

*I think that is actually the last example for today.*1526

*Thank you so much for joining us here at www.educator.com.*1528

*We will see you next time, bye.*1531

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