Question

What is the derivative of `f(x)=(cos x)^(sin x)`?

Answer 1

`f'(x) = cos(x)^sin(x)[cos(x)^{cos(x)}-sin^2x/cosx]`

Explanation:

Use logaritmic differentiation or rewrite as `f(x) = e^(ln(cosx)^sinx)`

I'll use logarithmic because I think it is easier to read.

`y=(cosx)^sinx`

`lny = ln(cosx)^sinx = sinxln(cosx)`

Diefferentiate implicitely:

`1/y dy/dx = cosxln(cosx)+sinx[-sinx/cosx]`

Solve for `dy/dx`

`dy/dx = (cosx)^sinx[cosxln(cosx)-sin^2x/cosx]`

Rewrite the answer to taste.