Question

How do you differentiate `f(x)=csce^(4x)` using the chain rule.?

Answer 1

`f'(x)=-4e^(4x)csc(e^(4x))cot(e^(4x))`

Explanation:

The chain rule states that when differentiating a function inside of a function, (1) differentiate the outside function and leave the inside function as is, and (2) multiply this by the derivative of the inside version.

In `f(x)=csc(e^(4x))`, we have the outside function `csc(u)` and the inside function `e^(4x)`.

Thus, since the derivative of `csc(x)` is `-csc(x)cot(x)`, we will have

`f'(x)=-csc(e^(4x))cot(e^(4x))*d/dx(e^(4x))`

Don't forget to multiply by the derivative of the inside function, which is `d/dx(e^(4x))`.

To differentiate `e^(4x)`, we will have to use the chain rule again.

The outside function is `e^u` and the inside function is `u=4x`. Since the derivative of `e^x` is still `e^x`, we have

`d/dx(e^(4x))=e^(4x)*d/dx(4x)`

Note that `d/dx(4x)=4`, so we know that

`d/dx(e^(4x))=4e^(4x)`

Plug this back in to the derivative of the whole function:

`f'(x)=-4e^(4x)csc(e^(4x))cot(e^(4x))`