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## AP®︎/College Calculus BC

### Course: AP®︎/College Calculus BC>Unit 3

Lesson 1: The chain rule: introduction

# Common chain rule misunderstandings

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) ?
• Wouldn't the answer be Tan (x)?
• `tan(x) = sin(x)/cos(x)`. This is `cos(x)/sin(x) = cot(x)`.
• 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?
• 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!
• When do you stop using the chain rule?
• When you get down to the very innermost function, which is just x.
• 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).
• you could definitely write it as cot(x). Both are right.
• 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?
• 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).
• Instead of 1/sin(x) times cos(x) for the final answer, could you not simplify it to cos(x)/sin(x)? Which also equals cot(x).