1. Given the function $f(x)=x^{\sin x}$, we want to find its derivative $f'(x)$ using logarithmic differentiation. 2. Start by taking the natural logarithm of both sides: $$\ln f(x) = \sin x \cdot \ln x$$ 3. Differentiate both sides with respect to $x$ using the product rule on the right-hand side and chain rule on the left: $$\frac{f'(x)}{f(x)} = \cos x \cdot \ln x + \sin x \cdot \frac{1}{x}$$ 4. Multiply both sides by $f(x) = x^{\sin x}$ to solve for $f'(x)$: $$f'(x) = x^{\sin x} \left( \cos x \ln x + \frac{\sin x}{x} \right)$$ 5. Simplify the fraction inside the parenthesis: $$f'(x) = x^{\sin x} \frac{x \cos x \ln x + \sin x}{x}$$ Final answer: $$\boxed{f'(x) = x^{\sin x} \frac{x \cos x \ln x + \sin x}{x}}$$