Question

The rate of change of \( x^{\sin x} \) with respect to \( (\sin x)^x \) is:

• A. \( \frac{x^{\sin x} \left(\frac{\sin x}{x} + \cos x \log x \right)}{(\sin x)^x (x \cot x + \log \sin x)} \)
• B. \( \frac{(\sin x)^x (x \cot x + \log \sin x)}{\sin x \left(\frac{\sin x}{x} + \cos x \log x \right)} \)
• C. \( y \left( \frac{\sin x}{x} + \cos x \log x \right) \)
• D. \( (\sin x)^x (x \cot x + \log \sin x) \)

Hint:

For functions of the form \( f(x)^{g(x)} \), take the natural logarithm first and then differentiate both sides using implicit differentiation.

Answer 1

The Correct Option is A

We need to determine: \[ \frac{d}{dx} \left( x^{\sin x} \right) \Bigg/ \frac{d}{dx} \left( (\sin x)^x \right). \] Step 1: Differentiating \( x^{\sin x} \)
Taking the natural logarithm: \[ y = x^{\sin x} \Rightarrow \ln y = \sin x \ln x. \] Differentiating both sides: \[ \frac{1}{y} \frac{dy}{dx} = \cos x \ln x + \sin x \cdot \frac{1}{x}. \] \[ \frac{dy}{dx} = x^{\sin x} \left( \cos x \ln x + \frac{\sin x}{x} \right). \] Step 2: Differentiating \( (\sin x)^x \)
Taking the natural logarithm: \[ z = (\sin x)^x \Rightarrow \ln z = x \ln \sin x. \] Differentiating both sides: \[ \frac{1}{z} \frac{dz}{dx} = \ln \sin x + x \cot x. \] \[ \frac{dz}{dx} = (\sin x)^x (x \cot x + \log \sin x). \] Step 3: Finding the rate of change
\[ \frac{\frac{dy}{dx}}{\frac{dz}{dx}} = \frac{x^{\sin x} \left(\frac{\sin x}{x} + \cos x \log x \right)}{(\sin x)^x (x \cot x + \log \sin x)}. \]