derivative of x^cosx

Question

i tried using a^x(lna) and got x^cosx(lnx) but I'm not sure if that's right or if I'm even done yet.

Answer 1

Rewrite the function as e ^{(ln x)(cos x)}

 

So by the chain rule, the derivative is:

 

e ^{ln x * cos x} D_x (ln x * cos x) = x ^{cos x} D_x (ln x * cos x)

 

The derivative of the second expression is ln x (-sin x) + (1/ x) cos x = cos x / x - ln x sin x

 

So the derivative of the entire function is:

 

x ^{cos x} (cos x / x - ln x sin x)