find the derivative of (cosx)^(sinx)^x

Question

Solve to find the derivative of the following function: (cosx)^(sinx)x

Do not simplify the answer.

Answer 1

Let $y=(\cos x)^{(\sin x)^x}$. Then $\ln y=(\sin x)^x\ln(\cos x)$. Before we continue, let's find the derivative of $(\sin x)^x$ in terms of $x$.

Let $z=(\sin x)^x$. Then $\ln z=x\ln(\sin x)$. Derive both sides in terms of $x$:

$$\frac{z'}{z}=\ln(\sin x)+x\frac{1}{\sin x}(\cos x)$$

$$z'=(\sin x)^x\left(\ln(\sin x)+\frac{x\cos x}{\sin x}\right)$$

We can now continue: derive both sides in terms of $x$:

$$\frac{y'}{y}=z'\ln(\cos x)+(\sin x)^x\frac{1}{\cos x}(- \sin x)$$

Now substitute $y$ and $z'$ with known values, simplify.