Main content

## AP®︎/College Calculus AB

### Unit 3: Lesson 1

The chain rule: introduction- Chain rule
- Common chain rule misunderstandings
- Chain rule
- Identifying composite functions
- Identify composite functions
- Worked example: Derivative of cos³(x) using the chain rule
- Worked example: Derivative of √(3x²-x) using the chain rule
- Worked example: Derivative of ln(√x) using the chain rule
- Chain rule intro

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Common chain rule misunderstandings

AP.CALC:

FUN‑3 (EU)

, FUN‑3.C (LO)

, FUN‑3.C.1 (EK)

Three common student misconceptions when applying the Chain Rule (from AP team at College Board).

## Want to join the conversation?

- How do we find the derivative of a logarithm? I don't remember learning that the derivative of ln(x) was 1/x. Also, couldn't the derivative be simplified to cot(x) since it is cos(x)/sin(x) ?(14 votes)
- This is covered here:

https://www.khanacademy.org/math/ap-calculus-ab/ab-derivatives-advanced/ab-diff-log/v/derivative-of-lnx`cot(x)`

would be an acceptable answer, but it is often easiest to leave answers in terms of the`sin`

and`cos`

(since few of us have the rules for manipulating all of the trig functions memorized, but do remember how to deal with`sin`

and`cos`

).(17 votes)

- Wouldn't the answer be Tan (x)?(1 vote)
`tan(x) = sin(x)/cos(x)`

. This is`cos(x)/sin(x) = cot(x)`

.(31 votes)

- When do you stop using the chain rule?(1 vote)
- When you get down to the very innermost function, which is just x.(9 votes)

- Instead of using the Chain Rule can't we use the rule applicable to logs:

F(X)=In(g(x))

F'(X)= g'(x)/g(x)

Therefore, using the example given:

f(x)= In(sin(x))

f'(x)= cos(x)/sin(x)

Is there anything wrong with using this method?(3 votes)- This method works. Note that the rule applicable to logs that you mentioned logically follows from the Chain Rule and the fact that the derivative of ln(x) is 1/x.

Have a blessed, wonderful new year!(2 votes)

- Is there a reason why the function in question doesn't multiply out to tan(x) in the end? Because sin(x) over cos(x) is equivalent to tan(x).(2 votes)
- you could definitely write it as cot(x). Both are right.(3 votes)

- couldn't we simplify the function even more by doing 1/sin(x)*cos(x)=cos(x)/sin(x)=1/tan(x)?(2 votes)
- i am confused d/dx of ln (x) is 1/x hence d/dx of Ln ( sinx ) is the the derivative of what inside the ( ) which is sin x over whats inside the ()

which means d/dx = cosx/sinx whith out the need of using chain rule , u can get the answer simply by applying the simple derivative rules , so why use the chain rule ??(2 votes) - Why is the very top equation permissible / valid?

By saying:

d/dx [ f(g(x)) ] = f'(g(x)) * g'(x)

Isn't he essentially saying that:

f'(g(x)) = f'(g(x)) * g'(x)

Which clearly doesn't make sense.

Perhaps it's an issue that with f'(x) notation, where you can't specify what the derivative's "with respect to" it?(2 votes)- The prime notation f'(x) assumes that we mean the derivative with respect to x. The left-hand-side in your first equation is the derivative with respect to x, the left-hand-side in your second equation is the derivative with respect to g(x).(1 vote)

- at5:42what would the actual value of f'(g'(x)) be?(1 vote)
- the transcript is off... is it just me? or is it also someone else?(1 vote)

## Video transcript

What we're gonna do in this video is focus on key misunderstandings
that folks often have, and we actually got
these misunderstandings from the folks who write the AP exams, from the actual College Board. So let's say that we are trying to take the derivative of the expression. So let's say we're taking the derivative of the expression, the natural log of sine of x. So the first key misconception
or misunderstanding that many people have is when you're dealing with transcendental functions like this, and transcendental functions
is just a fancy word for these functions like
trigonometric functions, logarithmic functions, that don't use standard
algebraic operations. But when you see transcendental
functions like this or compositions of them, many people confuse this with
the product of functions. So at first when they look at this, they might see this as being the same as the derivative with respect to x of natural log of x, natural log of x, times sine of x. And you can see just the
way that it's written, they look very similar, but this is the product of two functions. If you said natural log of x is f of x, and sine of x is g of x, this is the product of sine and g of x, sorry this is the product
of f of x and g of x, and here you would use the product rule. So to actually compute this, you would use the
product, the product rule. But this is a composition. Here you have f of g of x, not f of x times g of x. So here you have that is our g of x, it equals sine of x, and then our f of g of x is the natural log of sine of x. So this is f of g of x, f of g of x just like that. If someone asks you just what f of x was, well that would be natural log of x, but f of g of x is
natural log of our g of x, which is natural log of sine of x. So that's the key first thing, always make sure whether you're gonna use, especially with these
transcendental functions, that hey if this is a
composition you've gotta use the chain rule, not the product rule. It's not the product. Now sometimes you have a combination, you have a product of compositions, and then things get a
little bit more involved. But pay close attention to make sure that you're not dealing
with a composition. Now the next misconception students have is even if they recognize, okay I've gotta use the chain rule, sometimes it doesn't
go fully to completion. So let's continue using this example. The chain rule here says, look we have to take the
derivative of the outer function with respect to the inner function. So if I were to say, in this case, f of x is natural log of x, f of g of x is this expression here. So if I wanna do this first part, f prime of g of x, f prime of g of x, well the derivative of
the natural log of x is one over x. So the natural log,
derivative of natural log of x is one over x, but we don't want the
derivative where the input is x. We want the derivative
when the input is g of x. So instead of it being one over x, it's gonna be one over g of x. One over g of x, and we know that g of x
is equal to sine of x. That's equal to sine of x. Now one key misunderstanding
that the folks of the College Board told us about is many students stop right there. They just do this first part, and then they forget to
multiply this second part. So here we are not done. We need to take this and
multiply it times g prime of x. And let me write this down. g prime of x, what would that be? Well the derivative of sine
of x with respect to x, well that's just going to be cosine of x, cosine of x. So in this example right over here, the derivative is going to be, let's see if I can
squeeze it in over here, it's going to be one over
sine of x which is this part, times cosine of x. So let me write it down. It is going to be one over sine of x, we'll do that in that other color, one over sine of x, and then times cosine of x. So once again, just to make sure that you don't fall into one of these misconceptions. Let me box this off so it's a little bit, it's a little bit cleaner. So to just make sure that you don't fall into one of these misconceptions here, recognize the composition, that this is not the
product of natural log of x and sine of x. It's natural log of sine of x. And then when you're actually
applying the chain rule, derivative of the outside
with respect to the inside, so the derivative of natural
log of x is one over x, so that applied when the input is g of x is one over sine of x. And then multiply that
times the derivative of the inner function. So don't forget to do
this right over here. Now another misconception
that students have, is instead of doing what we just did, instead of applying the
chain rule like this, they take the derivative
of the outer function with respect to the derivative
of the inner function. So for example, they would compute this, f prime of g prime of x, f prime of g prime of x. Which in this case, f
prime of x is one over x, but if the input is g prime of x, g prime of x is cosine of x. So many students end up doing this where they take the
derivative of the outside, and they apply the input into that, they use the derivative
of the inside function. This is not right. Be very careful that
you're not doing that. You do the derivative
of the outside function with respect to the inside function, not taking it's derivative, and then multiply, don't
forget to multiply, times the derivative of
the inside function here. So hopefully that helps a little bit. If all of this looks
completely foreign to you, I encourage you to watch the whole series of chain rule introductory videos and worked examples we have. This is just a topping on top of that to make sure that you don't
fall into these misconceptions of applying the product rule when you really need to
be applying the chain rule or forgetting to do
part of the chain rule, multiplying by g prime of x, or evaluating f prime of g prime of x. So hopefully that helps.