What is $\frac{d\left( (\cos(x))^{\cos(x)}\right)}{dx}$? How would you work something like this out? 
Are there similar problems to
$$\frac{d\left( (\cos(x))^{\cos(x)}\right)}{dx}$$
which could be worked out the same way?

Hint: Given proper domain for the function so that $\cos(x) >0$ we can write:
$$(\cos x)^{\cos x} = e^{(\cos x) \ln(\cos x)}$$
