49–55. Derivatives of tower functions (or g^h) Find the derivative of each function and evaluate the derivative at the given value of a.
f (x) = (sin x)^In x; a = π/2

Verified step by step guidance

1. Step 1: Recognize that the function f(x) = (\sin x)^{\ln x} is a tower function of the form g(x)^{h(x)}. To differentiate it, use the logarithmic differentiation technique.
2. Step 2: Take the natural logarithm of both sides: \ln f(x) = \ln((\sin x)^{\ln x}) = \ln x \cdot \ln(\sin x).
3. Step 3: Differentiate both sides with respect to x. For the left side, use the chain rule: \frac{d}{dx}[\ln f(x)] = \frac{1}{f(x)} \cdot f'(x). For the right side, use the product rule: \frac{d}{dx}[\ln x \cdot \ln(\sin x)] = \ln(\sin x) \cdot \frac{1}{x} + \ln x \cdot \frac{1}{\sin x} \cdot \cos x.
4. Step 4: Solve for f'(x) by multiplying both sides by f(x): f'(x) = f(x) \cdot \left(\frac{\ln(\sin x)}{x} + \ln x \cdot \cot x\right).
5. Step 5: Evaluate f'(x) at x = \frac{\pi}{2}. Substitute x = \frac{\pi}{2} into the expression for f'(x) and simplify, noting that \sin(\frac{\pi}{2}) = 1 and \ln(1) = 0.