What is f'(x) if f(x)=x^(sin x)?

First thing, we'll take natural logarithms both
sides:

ln f(x) = ln [x^(sin
x)]

We'll apply the power rule of
logarithms:

ln f(x) = sin x* ln
x

We'll differentiate with respect to x both
sides:

f'(x)/f(x) = (sin x*ln
x)'

We'll apply product rule to the right
side:

f'(x)/f(x) = cos x* ln x + (sin
x)/x

Now, we'll multiply both sides by
f(x):

f'(x) = f(x)*[cos x* ln x + (sin
x)/x]

But f(x) = x^(sin x), therefore f'(x) =
[x^(sin x)]*[cos x* ln x + (sin x)/x].