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) = (4 sin x+2)^cos x; a = π

Verified step by step guidance

1. Identify the function f(x) = (4 sin x + 2)^(cos x) and recognize that it is a tower function, which requires the use of logarithmic differentiation to find the derivative.
2. Take the natural logarithm of both sides: ln(f(x)) = cos(x) * ln(4 sin x + 2). This simplifies the differentiation process.
3. Differentiate both sides with respect to x using the chain rule on the left side and the product rule on the right side: (1/f(x)) * f'(x) = -sin(x) * ln(4 sin x + 2) + cos(x) * (4 cos x)/(4 sin x + 2).
4. Multiply both sides by f(x) to isolate f'(x): f'(x) = f(x) * [-sin(x) * ln(4 sin x + 2) + cos(x) * (4 cos x)/(4 sin x + 2)].
5. Substitute x = π into the expression for f'(x) to evaluate the derivative at the given value.