find the derivative of the function and evaluate the derivative at the given value of a.\n\nf(x)=(sin…

find the derivative of the function and evaluate the derivative at the given value of a.\n\nf(x)=(sin x)^{\\ln 4x};a = \\frac{\\pi}{2}\n\nfind the derivative of the function.\n\n\\frac{d}{dx}(sin x)^{\\ln 4x}=(sin x)^{\\ln 4x}(\\frac{\\ln sin x}{x}+\\ln 4x\\cdot cot x)\n\n(use parentheses to clearly denote the argument of each function.)\n\nevaluate the derivative at the given value of a.\n\nf(a)=\\square (type an exact answer, using \\pi as needed.)

find the derivative of the function and evaluate the derivative at the given value of a.\n\nf(x)=(sin x)^{\\ln 4x};a = \\frac{\\pi}{2}\n\nfind the derivative of the function.\n\n\\frac{d}{dx}(sin x)^{\\ln 4x}=(sin x)^{\\ln 4x}(\\frac{\\ln sin x}{x}+\\ln 4x\\cdot cot x)\n\n(use parentheses to clearly denote the argument of each function.)\n\nevaluate the derivative at the given value of a.\n\nf(a)=\\square (type an exact answer, using \\pi as needed.)

Answer

Explanation:

Step1: Substitute (x = a=\frac{\pi}{2}) into the derivative formula

We have (f^{\prime}(x)=(\sin x)^{\ln 4x}\left(\frac{\ln\sin x}{x}+\ln 4x\cdot\cot x\right)). When (x = \frac{\pi}{2}), (\sin\left(\frac{\pi}{2}\right)=1), (\cot\left(\frac{\pi}{2}\right)=\frac{\cos\left(\frac{\pi}{2}\right)}{\sin\left(\frac{\pi}{2}\right)} = 0), and (\ln\left(4\times\frac{\pi}{2}\right)=\ln(2\pi)).

Step2: Calculate each part of the formula

  • For the first - factor ((\sin x)^{\ln 4x}), when (x=\frac{\pi}{2}), ((\sin\frac{\pi}{2})^{\ln(2\pi)}=1^{\ln(2\pi)} = 1).
  • For the second - factor (\frac{\ln\sin x}{x}+\ln 4x\cdot\cot x), substitute (x = \frac{\pi}{2}):
    • (\frac{\ln\sin\left(\frac{\pi}{2}\right)}{\frac{\pi}{2}}+\ln(2\pi)\cdot\cot\left(\frac{\pi}{2}\right))
    • Since (\sin\left(\frac{\pi}{2}\right) = 1), (\ln\sin\left(\frac{\pi}{2}\right)=\ln(1) = 0), and (\cot\left(\frac{\pi}{2}\right)=0), the second - factor is (\frac{0}{\frac{\pi}{2}}+\ln(2\pi)\times0=0).

Answer:

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