What is the derivative of $f (x) = {(cos x)}^{sin x}$ $f(x)=(cos x)^(sin x)$ ?

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Answer 1 · 2015-11-09T02:31:17

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

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.

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