Question

How do you differentiate `f(x)=cot(1/e^x)` using the chain rule?

Answer 1

`1/e^(x)csc^2(1/e^x)`

Explanation:

To do this we need the chain rule.

First we find the derivative of the function inside the cot function. Remember `1/e^x=e^-x`

`d/dx(e^-x)=-e^(-x)=-1/e^(x)`

Also the derivative of `cot(x)`:
`d/dxcot(x)=-csc^2(x)`

The chain rule states that: `d/dxf(g(x))=g'(x)f'(g(x))`
So we take the derivative of the inside and multiply it by the derivative of the outside. We found the derivatives, we just need to multiply to get:

`1/e^(x)csc^2(1/e^x)`