Find $f^{\prime}(x)$ if $f^{\prime}(x)=(\sin x)^{\sin x}$ for all $0<x<\pi$.

Step 1: Take the natural logarithm of both sides

To differentiate a function of the form $u(x)^{v(x)}$, it is often helpful to use logarithmic differentiation. We start by taking the natural logarithm of both sides of the equation.

Step 2: Apply logarithm property

Using the logarithm property $\ln(a^b) = b \ln a$, we can simplify the right-hand side of the equation.

Step 3: Differentiate both sides with respect to x

Now, we differentiate both sides of the equation with respect to $x$. On the left side, we use the chain rule. On the right side, we will use the product rule.

Step 4: Apply product rule and chain rule

Applying the product rule $\frac{d}{dx}(uv) = u'v + uv'$ to the right side, where $u = \sin x$ and $v = \ln(\sin x)$. The derivative of $\sin x$ is $\cos x$, and the derivative of $\ln(\sin x)$ is $\frac{1}{\sin x} \cdot \cos x$ by the chain rule.

Step 5: Simplify the expression

We simplify the right-hand side by canceling out $\sin x$ in the second term.

Step 6: Solve for $f^{\prime}(x)$

Finally, we multiply both sides by $f(x)$ and substitute back $f(x) = (\sin x)^{\sin x}$. We can also factor out $\cos x$ from the expression.

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