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 of the Trigonometric Functions

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
- Derivatives of the Trigonometric Functions
- Let's Find the Derivative of f(x) = sin x
- Important Limits to Know
- d/dx (sin x)
- d/dx (cos x)
- d/dx (tan x)
- d/dx (csc x)
- d/dx (sec x)
- d/dx (cot x)
- Example I: Differentiate f(x) = x² - 4 cos x
- Example II: Differentiate f(x) = x⁵ tan x
- Example III: Differentiate f(x) = (cos x) / (3 + sin x)
- Example IV: Differentiate f(x) = e^x / (tan x - sec x)
- Example V: Differentiate f(x) = (csc x - 4) / (cot x)
- Example VI: Find an Equation of the Tangent Line
- Example VII: For What Values of x Does the Graph of the Function x + 3 cos x Have a Horizontal Tangent?
- Example VIII: Ladder Problem
- Example IX: Evaluate
- Example X: Evaluate

- Intro 0:00
- Derivatives of the Trigonometric Functions 0:09
- Let's Find the Derivative of f(x) = sin x
- Important Limits to Know
- d/dx (sin x)
- d/dx (cos x)
- d/dx (tan x)
- d/dx (csc x)
- d/dx (sec x)
- d/dx (cot x)
- Example I: Differentiate f(x) = x² - 4 cos x 7:56
- Example II: Differentiate f(x) = x⁵ tan x 9:04
- Example III: Differentiate f(x) = (cos x) / (3 + sin x) 10:56
- Example IV: Differentiate f(x) = e^x / (tan x - sec x) 14:06
- Example V: Differentiate f(x) = (csc x - 4) / (cot x) 15:37
- Example VI: Find an Equation of the Tangent Line 21:48
- Example VII: For What Values of x Does the Graph of the Function x + 3 cos x Have a Horizontal Tangent? 25:17
- Example VIII: Ladder Problem 28:23
- Example IX: Evaluate 33:22
- Example X: Evaluate 36:38

### AP Calculus AB Online Prep Course

### Transcription: Derivatives of the Trigonometric Functions

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

*Today, we are going to be talking about the derivatives of the trigonometric functions.*0004

*Let us jump right on in.*0008

*Let us start off by finding the derivative of sin x.*0010

*Let me see, I will stick with blue today.*0016

*Let us find the derivative of f(x) = the sin(x).*0024

*We have the derivative definition.*0040

*It is going to be the limit, as h approaches 0 of f(x + h - f(x)/ h).*0045

*We are going to form quotient and I take the limit as h goes to 0.*0055

*I’m just going to put little quotation marks here for the limit h approaches 0.*0061

*I do not want to keep writing it, I hope you will forgive.*0064

*Our function is sin(x).*0068

*We have sin(x) + h – sin(x)/ h.*0070

*And then, when we expand this sin(x) + h, you remember the addition formulas*0080

*or you remember having seen the addition formulas, it becomes sin x cos h + cos x sin h + sin x, we have all of that /h.*0086

*What I’m going to do is I’m going to separate these out.*0104

*I’m going to combine a couple of these.*0108

*Let me go ahead and put a.*0111

*This is the limit as h approaches 0 of, I’m making myself a little more room here.*0112

*We are going to have sin x cos h – sin x/ h + this second part which is cos x sin h/ h.*0120

*This is going to be, this is going to equal that.*0144

*This is going to equal the limit as h approaches 0.*0147

*Let me see, I’m going to take out the sin x.*0152

*The sin x × this is going to be cos h -1/ h.*0156

*I’m keeping the x and h together.*0166

*That is all I’m doing here.*0168

*+ cos x × sin h/ h.*0170

*That is it, I just put things together so that I can actually do it.*0177

*Let me go ahead and write it here.*0183

*In this case, this is the limit as h approaches 0.*0184

*This × this + this × this.*0188

*It is going to be the limit of this × the limit of this + the limit of this × the limit of this.*0191

*Limit theorems, let me just go ahead and take them one at a time.*0198

*This actually equal the limit as h approaches 0 of sin x × the limit as h approaches 0 of cos h -1/ h*0202

*+ the limit as h approaches 0 of cos x × the limit as h approaches 0 of sin h/ h.*0220

*Here is what happens.*0234

*As h goes to 0, notice there is no h in here.*0235

*The limit as h approaches 0 of sin x is just sin x.*0238

*This is just sin x.*0244

*This limit right here, let me do this in red.*0248

*These two limits, this limit and this limit are very important limits.*0253

*I’m not going to demonstrate how they end up being what they equal but these are two limits that you definitely have to know.*0258

*This one is going to be 0.*0266

*It is sin x × 0 + the limit as h goes to 0 of cos x is just cos x.*0268

*This limit right here, the limit of the sin of h/ h as h goes to 0 is actually equal to 1.*0275

*Let us see, therefore, the derivative of sin x = cos x, this term goes to 0.*0285

*Once again, these two limits are very important to know.*0299

*You just have to know them, very important to know.*0309

*First one, the limit as x approaches 0 of the sin(x)/ x that is equal to 1.*0320

*The other limit, the limit as x approaches 0 of cos x -/ x that is equal to 0.*0332

*We will be using these limits to evaluate other limits, until the time that we learn something called L’Hospitals rule.*0348

*And then, we would not have to worry about these limits anymore.*0355

*But again, they are very important limits, they do tend to come up.*0357

*As far as the formulas are concerned, let us go back to black here.*0361

*The derivative with respect to x of the sin(x) = the cos of x.*0371

*Going with the same process of the different quotient, taking the limit.*0378

*Again, we will not go through the pool for each one, we are just going to list them because again,*0381

*what we want to do is we just want to develop technique.*0385

*We want to be able to differentiate quickly.*0390

*In this part of the course, this is all we are doing, finding ways to differentiate all this different functions.*0392

*The derivative with respect to x of cos x is equal to -sin x.*0399

*Make sure you watch your sign.*0406

*Derivative of sin is cos, derivative of cos is –sin.*0407

*The derivative of the tangent function, the tan of x = sec² x.*0414

*The derivative of cosec of x = -cosec x cot x.*0424

*The derivative with respect to x of sec x is equal to sec x tan x.*0436

*The derivative of the cotangent function, cot x is equal to –cosec² x.*0448

*These are our 6 trigonometric functions and their derivatives.*0456

*The rest is now just examples.*0460

*Let us jump right onto the examples and see what we can do.*0464

*Again, really straightforward, nothing particularly complicated.*0467

*Just a question of memorizing via derivative formulas and doing a bunch of problems.*0471

*Differentiate f(x) = x² – cos(x), very simple.*0478

*F’(x), if you want you can use the dy dx symbolism as well, it does not really matter.*0484

*You are going to see both.*0496

*The derivative of this is 2x 4 cos x, it is going to be -, this – stays and this is going to be -4 sin x.*0500

*The 4 is just a constant.*0513

*Let us go ahead and take the constant to be consistent.*0516

*-4 × the derivative of cos x which is –sin x.*0520

*Our f’(x) is equal to 2x, - and -, +4 sin x.*0528

*There you go, nice and straightforward.*0536

*You just have to use the differentiation formulas.*0538

*All we are doing is practice, practice, practice.*0541

*Example 2, differentiate f(x) = x⁵ tan x.*0546

*Here we have a product rule.*0551

*This is for our first function, this is our second function.*0553

*We have product rule, in this case one of them is a monomial and the other one is a trigonometric function.*0556

*It is going to be the derivative of this × that + the derivative of this × that.*0563

*F’(x) is equal to the derivative of this is 5x⁴ × this × the tan(x) + the derivative of this × that + sec² x × x⁵.*0569

*We just want to go ahead and bring this x⁵ in front.*0593

*You do not have to, you can leave it like this, it is not a problem.*0596

*Traditionally, we tend to bring everything forward and leave the trigonometric function at the end.*0599

*We have 5x⁴ tan(x) + x⁵ sec² x.*0604

*That is it, nice and simple.*0614

*Do you want to simplify it more, do you want to express it in terms of sin and cos, you can if you want to.*0618

*Simplification is really going to be, it is a totally individual thing how far you want to simplify.*0623

*You remember from product rule, quotient rule, in the previous lessons.*0629

*Where do you stop, at some point, you just have to stop.*0634

*Ultimately, it does not really matter where you stop.*0638

*You want to simplify as much as reasonable but it is going to be subjective.*0640

*What is reasonable to you might not be reasonable to somebody else.*0645

*Ultimately, you just have to ask what is reasonable to your teacher.*0648

*That takes care of that one.*0653

*Differentiate x(x) = cos x/3 + sin x.*0657

*Now we have a quotient rule and some trigonometric functions.*0661

*You remember the quotient rule, if you have f/g and if you take the derivative, you are going to have g f’ – f g’/ g².*0665

*This is f and this is g, f is cos x, 3 + sin x is g.*0680

*Therefore, we have f’(x) is equal to this × the derivative of that.*0685

*We have 3 + sin x × the derivative of cos x which is –sin x, - this cos x × the derivative of this.*0691

*The derivative of this is 0 +, this is just cos x/ 3 + sin x².*0705

*Let us go ahead and distribute this out.*0719

*Let us see.*0724

*You know what, I think I made a mistake on my paper so I will do it here, just as is.*0732

*Sin x cancels, this is going to be -3 sin x – sin² x – cos² x/ 3 + sin x².*0736

*Let me see, I have got -3 sin x.*0762

*I’m going to write this part -.*0767

*I’m going to write this as sin² x + cos² x.*0772

*The – distributes giving me the two negatives, /3 + sin x².*0778

*This right here is the identity.*0790

*This is just 1.*0793

*Our final answer -3 sin x -1/ 3 + sin x².*0797

*Let us go ahead and factor out that negative.*0814

*Let us write it as -3 sin x + 1.*0819

*No, that is not going to simplify anything.*0830

*I thought maybe that something might cancel but it is not.*0832

*You know what, we can stop there, not a problem.*0835

*There you go, quotient rule, trigonometric functions.*0839

*Reasonable, very straightforward.*0843

*Now we have differentiate e ⁺x tan x – sec x.*0848

*For these two, we have product rule.*0855

*We have e ⁺x and tan x – sec x, we just differentiate straight.*0857

*F’(x) = this × the derivative of that = e ⁺x × sec² x + the derivative of this × that.*0862

*I will leave it in the front.*0877

*It is the derivative of this × that + the derivative of,*0883

*Wait a minute, what am I doing here, let us start again.*0889

*The derivative of this × that.*0895

*We have e ⁺x × tan x, there we go, + the derivative of this × that +, I will leave this in front, e ⁺x sec² x.*0896

*And then, -the derivative of the sec x which is sec x tan x.*0912

*That is fine, you can go ahead and leave it like that.*0923

*You have a tan x tan x, sec² x sec² x, e ⁺x.*0926

*You might as well leave it like that, I do not think it will make it any easier.*0931

*That was nice and simple, good.*0935

*Again, we have another quotient rule.*0940

*Let us see what we can do with this one.*0943

*F'(x) is equal to this × the denominator × the derivative of the numerator.*0947

*We have cot(x) × the derivative of this which is going to be -cosec x cot x – this*0954

*which is cosec x – 4 × the derivative of the cot x which is –cosec² x/ the cot² (x).*0971

*If you want you can write this cot(x)², not a problem.*0990

*It is just we tend to put these squared on the cot.*0996

*Whatever is easier for you, you will see in a little bit that I actually do write that sometimes.*0999

*In the next lesson, when we talk about something called the chain rule, it is often easier.*1006

*At least in the beginning, when you are getting used to it.*1011

*To write something like the cos² (x), it is actually easier to write it as cos(x)².*1013

*It tends to make the differentiation a little bit easier.*1021

*Again, as you sort of get accustomed to the idea of seeing the squared here.*1023

*Let us see what we can do with this.*1029

*Let us go ahead and multiply this out, I guess the numerator.*1030

*We have –cosec x cot² x.*1037

*This is – × -, we have + cosec³ x.*1046

*This is – × - × -, it stays -4 cosec² x/ cot² x.*1056

*cosec and cosec, I’m going to pull out a cosec.*1072

*-cosec x × cot² x – cosec² x.*1074

*The – and – gives me the +.*1098

*And then, this is going to be a -, so it is going to be a +.*1101

*-cosec, that is going to be -, it is going to be +.*1110

*This is -, that is -, this is going to be +4 cosec² x.*1114

*It is always not that easy to keep track of all these signs and things.*1122

*By all means, if I make a mistake, it is definitely going to happen.*1125

*That is the nature of the game.*1129

*You have all these symbols floating around.*1131

*Mistakes are going to happen.*1133

*/cot² x.*1135

*Let us see, I think I got this right.*1141

*Let me double check.*1143

*-cosec, I pulled out a negative.*1144

*This becomes -, - × - is + goes to +.*1147

*This is -, so -.*1151

*Yes, I think we are good.*1152

*This becomes, I’m going to write this –cosec as -1/ sin(x).*1155

*Here, cot² – cosec², if you remember from your trigonometric identities, it is equal to 1.*1162

*This is going to be 1 + 4 cosec² x.*1169

*The cot² x, I’m going to write as cos² x/ sin² x.*1177

*When I do that, I’m going to go ahead and cancel this sign and one of these signs.*1186

*Let us see, that is going to give me.*1193

*Now I have f’(x) is equal to,*1201

*Let me write that part over, actually.*1209

*-1/ sin(x) × 1 + 4 cosec(x).*1211

*I think that was what is on top.*1220

*Here we have cos² x/ sin² x.*1222

*We cancel the sin and we can flip this.*1226

*This sin ends up coming to the top as sin x, the – stays.*1230

*Here we have 1 + 4 cosec x.*1237

*Over here, we have cos² x stays on the bottom.*1243

*Then, let me write this as sin(x) × 1 + 4/ cosec is sin(x)/ cos² (x).*1254

*Again, you do not really need to simplify all this much.*1275

*But to solve the identities, let us take it as far as we can take it.*1277

*This is going to be –sin(x), when I distribute this.*1284

*This, the sin x are going to cancel so it is going to be -4/ cos² (x).*1289

*I think that should take care of it.*1298

*I think that is as far as we want to take that one.*1303

*There you go.*1307

*Find an equation of the line tangent to y = x² sin² x at x = π/6.*1310

*We are looking for an equation of the tangent line.*1318

*We are definitely going to be using y – y1 = m × x – x1.*1321

*We have x, that is π/6, it is going to be our x1.*1331

*Let us go ahead and find the y1 first.*1335

*Y(π/6) is going to equal what we just put in there.*1339

*It is going to be π/6² × sin(π/6).*1346

*π/6² is π²/36 × sin(π/6), sin(30 degrees) is 0.5 or ½.*1352

*We end up with π²/72.*1363

*We know our point.*1370

*Now, π/6 and π²/72, that is tan.*1372

*Tan which means find the derivative.*1383

*Y’, this is a product rule.*1386

*It is going to be the derivative of this × this.*1391

*We get 2x × sin(x) + the derivative of this × this + cos(x) × x².*1393

*Y’ at π/6, we are trying to find the slope.*1406

*It is going to equal to × π/6 × sin(π/6) + cos(π/6) × π/6².*1414

*What do we have here?*1433

*Let us just go ahead and write it all out, it is not a problem.*1436

*This is going to be, let me do it over here.*1440

*2 × π/6, sin(π/6) is ½, cos(π/6) is √3/2, × π²/ 36.*1443

*Here we have, the 2 and 2 cancels.*1458

*We are left with π/6 + π² √3/72.*1460

*This is going to be our slope.*1473

*That is our slope.*1477

*Now we put it into the equation.*1481

*We get y – y1 which is π²/72 = the slope which is π/6 + π² √3/72 × x – π/6.*1484

*There you go, that is your equation.*1505

*Again, depending on the extent to which your teacher wants you to simplify that, you can go ahead and do so.*1509

*Nice, nothing particularly strange.*1515

*For what values of x does the graph of the function x + 3 cos(x) have a horizontal tangent?*1520

*Horizontal tangent means that the derivative is 0.*1527

*Horizontal tangent that implies that f’(x) is equal to 0.*1536

*Let us go ahead and take the derivative of this.*1544

*If f(x) = x + 3 × cos(x), that means that f’(x) is going to equal 1.*1548

*The derivative of cos(x) is – sin x, we have -3 sin x.*1560

*We are going to set that equal to 0.*1566

*1 – 3 sin x = 0, that means that 3 sin(x) is equal to 1.*1569

*That means sin(x) is equal to 1/3.*1577

*Therefore, x is equal to the inv sin(1/3).*1581

*This is a positive number.*1588

*The sin is positive in two quadrants.*1592

*We are going to get two answers.*1594

*We are going to get an angle of this quadrant.*1595

*We are going to get an angle of this quadrant.*1597

*The angle that we are taking, the first one is going to be this one.*1599

*The second one is going to be that one.*1602

*When I do this, when I put this into the calculator, I get x = in radians, 0.3398 radians.*1607

*That is this first angle.*1619

*The other angle, that is going to be 0.3398, that is going to be this angle over here but we do not measure it from here.*1623

*We take all of our measurements from the +x axis.*1630

*We are going to take π which is 180 - 0.3398.*1632

*π - 0.3398 rad.*1638

*There you go, once again, we draw this out.*1650

*This angle is x1, the 0.3398 it happens to be 19.47°.*1653

*The other one s 19.47°, in the second quadrant but we measured it from here.*1666

*This is x2, this is the π - 0.3398.*1675

*It happens to equal what is going to be 180 - 19.47.*1680

*We have got 160.52°.*1684

*There you, these are our two answers.*1689

*Of course, every 2π after that, it comes around again.*1693

*It is a periodic function.*1699

*Let us see, word problem, nice.*1704

*A ladder 12ft long rest against the wall, if the bottom of the ladder slides from the wall,*1710

*how fast does the height of the ladder against the wall change with respect to the angle that the ladder makes with the wall,*1715

*as the ladder is being pulled away at the base?*1723

*Let us go ahead and draw a picture here.*1726

*We have a wall here and we have our ladder.*1729

*Let me just go ahead.*1736

*We have our ladder that way.*1740

*They want to know, this ladder is 12ft long.*1742

*The bottom of the ladder slides away.*1747

*Now, this ladder is going to be moving this way which means that the top of the ladder is going to be moving that way.*1748

*How fast does the height of the ladder against the wall change?*1755

*How fast does this height changing?*1758

*Let us call this height y, let us call this x.*1761

*With respect to the angle that the ladder makes with the wall.*1766

*The angle that the ladder makes with the wall, let us call it θ.*1769

*They want to know how fast does the height of the ladder against the wall change, with respect to the angle that the ladder makes.*1776

*They want to know dy dθ.*1782

*They want to know how fast this changes, y with respect to the change in the angle θ.*1788

*That is dy dθ, that is it.*1797

*What that means is we need to find an equation y = some function of θ and we need to differentiate it, y’.*1802

*Y’ is dy dθ.*1813

*If this is what you are looking for, you need to find an equation of this equal to some function of this.*1819

*And then, you differentiate it and that will give you what this is.*1825

*You have a relationship here between y and θ and 12.*1829

*We have that the cos(θ) is equal to 1/12 which means that y = 12 × cos(θ).*1834

*Y’ which is equal to dy dθ is equal to -12 × cos(θ).*1850

*We are done, we have what we wanted.*1860

*The rate of change of y is equal to -12 × cos of whatever θ happens to be at that moment.*1865

*If θ happens to be some number that means y is changing.*1874

*Notice, it is negative, it s getting smaller.*1880

*The ladder is falling, it is going this way.*1884

*This negative sign confirms the fact that it is getting smaller.*1886

*If I were to do dx dθ, I would find that this is actually positive because it is getting bigger.*1891

* All of this is confirmed.*1897

* Once again, we are looking for dy dθ.*1899

*How fast does y change, when I change θ?*1902

*That means I have to find an equation of this, in terms of this, y as a function of θ.*1905

*I found it, now I just differentiated it and I have what I want.*1911

*Nice and simple.*1915

*If they gave us a specific θ, you would put it in and solve.*1916

*For example, if they said π/3.*1920

*Cos(π/3) is ½ so -12 × ½, it is going to be -6.*1924

*The rate of change of y when θ actually hits π/3, it is going to be changing by 6ft/s, something like that.*1932

*There you go, very important set of problems, this whole idea of the rate of change.*1941

*That is what we are doing, the rate of change.*1951

*These word problems in calculus are going to be profoundly important.*1952

*Again, the difficult part is not the actual differentiation, that is the easiest part of the whole problem.*1958

*The difficulty with these word problems is going to be taking a look at this and constructing what is going on,*1963

*and extracting an equation among the variables that you are interested in.*1968

*Keeping track of what it is that you want.*1974

*Very important.*1976

*When they say how fast does the height of the ladder against the wall change with respect to the angle?*1978

*That is dh dθ or in this case I chose y as my height, dy dθ.*1985

*I need an equation y, in terms of θ, that is just automatic.*1991

*That is what is going on, this is what you want to extract from these word problems.*1995

*Let us take a look at example number 9.*2003

*Evaluate the limit as x approaches 0 of tan(7x)/x.*2005

*Again, now we are going to use those limits that we saw.*2012

*The limit of sin x/ x and limit of cos x – 1/x.*2015

*The limit as x goes to 0.*2020

*We are going to use them to evaluate these limits.*2021

*Let us see what we can do.*2025

*In this particular case, we have 7x/ x.*2027

*This is not tan(x), this not sin(x)/x.*2029

*We need to sort of fiddle with this.*2034

*Let us see what we can do.*2037

*Let us see if we can somehow transform this and turn it into something that looks like something we already know.*2039

*What I'm going to do, I’m going to work with tan(7x)/x.*2047

*I’m going to leave the limit off for the time being.*2052

*I will come back to it at the end, when I simplify it.*2055

*I’m going to multiply this by 7/7.*2058

*I’m just multiplying by 1.*2060

*I’m not actually changing anything.*2061

*This equals 7 × tan(7x)/ 7x.*2064

*This is nice, now I have the argument of the trigonometric function and the denominator the same.*2072

*It starts to look interesting.*2077

*I’m going to express tan(7x), this is 7.*2079

*The tan(7x) is equal to sin(7x)/ cos(7x).*2088

*I’m going to write this as 7 × sin(7x)/ 7x × cos(7x).*2097

*That is it, all I have done is change the tangent to this.*2107

*I’m going to take this and this, and then this, separately.*2110

*This is equal to 7 × sin(7x)/ 7x × 1/ cos(7x).*2120

*Now I’m going to take the limit of that.*2133

*The limit as x approaches 0 of this thing 7 × sin(7x)/ 7x × 1/ cos(7x).*2136

*The limit as x approaches 0 of 7 is just 7.*2150

*The limit as x approaches 0 of sin(7x)/7x that is where we use that limit that we have.*2153

*The limit as x approaches 0 of sin(a)/a is equal to 1.*2160

*Here, as x approaches 0, cos(7) × 0 is cos(0).*2168

*Cos(0) is 1, 1/1 is 1.*2173

*Our final answer is 7.*2176

*I have transformed this thing, tan(7x)/ x by just manipulation into something that I actually recognize, using what is able to find that.*2178

*This limit is easy, this limit is easy, I get 7.*2191

*I hope that make sense.*2195

*Evaluate the limit as x approaches infinity, sin(1)/ x.*2200

*Let us take this x × sin(1)/ x.*2208

*We have x × sin(1)/ x.*2220

*I’m going to multiply by 1, in the form of 1/x/ 1/x.*2224

*That is equal to 1/x × x × sin(1)/x / 1/x.*2233

*I will just multiply 1/x and 1/x down below.*2248

*This and this cancel.*2253

*I’m left with the sin(1)/x/ 1/x.*2254

*I have this thing.*2261

*Now we take the limit, but this time the limit is saying as x goes to infinity of this thing.*2269

*I do not like doing that.*2278

*Sin(1)/x/ 1/x, I like it to be vertical.*2279

*This and this = a.*2286

*As x goes to infinity, as this goes to infinity, this whole thing goes to 0.*2293

*A is going to 0.*2301

*This is the same as the limit as x goes to 0 of sin(a)/a.*2304

*We know what this limit is already, it is equal to 1.*2317

*That is it, mathematical manipulation.*2320

* The question that most kids ask is that how do you know how to do this?*2326

*You do not, there is no way of looking at a problem and knowing exactly what steps you are going to take.*2329

*You just try.*2335

*What you saw here is the finished product.*2337

*What you did not see was the 5 or 6 different things that we try to do,*2341

*manipulations that did not go anywhere, that we ended up hitting a wall.*2346

*That is what is going to happen.*2350

*What you want to do in calculus from this point forward in your math work and your science work,*2352

*you want to disabuse yourself of this notion that you are supposed to look at a problem*2359

*and just all of the sudden know exactly what to do.*2364

*There are too many steps at this point in higher math and higher science,*2366

*for you to actually be able to see the entire path in your head.*2371

*Sometimes you will but sometimes you just have to start.*2374

*Especially, when it involves some form of mathematical manipulation to make the problem a little bit more practical.*2377

*You just have to start and you see where you go.*2382

*Maybe you would have tried x/x, that did not do anything.*2384

*You just try and if that does not work, if you hit a wall, you go back and you start again.*2388

*In this case, it turned out to be 1/x and 1/x.*2393

*It fell out, it fell out, perfect.*2396

*You get your answer that way.*2398

*Trust the mathematics.*2400

*You want to trust the mathematics and you want to trust your intuition.*2404

*But do not trust the fact that just because you do not see the entire path, that you know how to solve the problem.*2408

*You need to be able to work a couple of steps at a time because the solution might be so far ahead that you cannot actually see it.*2415

*The solution might be, there might be a curve, it might be along the curve.*2422

*You cannot see the curve ahead but if you take a couple of steps toward it, then you can see the curve ahead.*2426

*That is how this works.*2431

*Just do that, do not worry about it.*2434

*It is not like we look at these problems and automatically know what to do.*2437

*Most of the time, we do not, particularly with the things that we deal within our research or anything else.*2441

*We do not know, when we try things, most of the time we actually fail.*2447

*What you are seeing is the actual define of success.*2452

*It makes it look like this is really simple, let us just do this, no.*2456

*Anyway, I hope that helps.*2461

*Take good care and I will see you next time.*2464

*Thank you for joining us here at www.educator.com, bye.*2466

1 answer

Last reply by: Professor Hovasapian

Wed Jun 28, 2017 9:26 PM

Post by Peter Fraser on June 28 at 03:51:45 PM

Example X:

I got the same result by doing: y = x.sin(1/x); so y' = 1.cos(1/x); lim x --> inf so y' = 1.cos(0) = 1.1 = 1. Hope that's right :)

1 answer

Last reply by: Professor Hovasapian

Fri Mar 25, 2016 11:11 PM

Post by Avijit Singh on March 6, 2016

Hi Prof Raffi,

I was wondering if it is necessary to simplify (with the trig identities) once we have differentiated y. For example, in example 5 you were able to simplify quite a lot.

Will I lose marks on the AP exam if i was to leave it un-simplified? Thanks.

1 answer

Last reply by: Professor Hovasapian

Wed Jan 13, 2016 12:01 AM

Post by Sohan Mugi on January 10, 2016

Hello Professor Hovasapian. For the ladder problem(Example viii), wouldn't the derivative equal -12sin(theta)? Because the derivative of cos=-sin, so it would be -12sin(theta) right? Thank you.