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

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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.

Step 2: Take the natural logarithm of both sides: \(\ln\) f(x) = \(\ln\)((\(\sin\) x)^{\(\ln\) x}) = \(\ln\) x \(\cdot\) \(\ln\)(\(\sin\) x).

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.

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\(\right\)).

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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Chain Rule

The Chain Rule is a fundamental differentiation technique used to find the derivative of composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule is essential for differentiating functions like f(x) = (sin x)^(ln x), where both the base and the exponent are functions of x.

Logarithmic Differentiation

Logarithmic Differentiation is a method used to differentiate functions of the form y = f(x)^(g(x)), where both the base and the exponent are functions of x. By taking the natural logarithm of both sides, we can simplify the differentiation process, allowing us to use the properties of logarithms to bring down exponents and make the function easier to differentiate. This technique is particularly useful for functions like f(x) = (sin x)^(ln x).

Evaluating Derivatives at Specific Points

Evaluating derivatives at specific points involves substituting a given value into the derivative function after it has been computed. This process allows us to find the slope of the tangent line to the function at that particular point. In this case, after finding the derivative of f(x) = (sin x)^(ln x), we will substitute a = π/2 to determine the behavior of the function at that specific x-value.

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