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
\[ \frac{\frac{dy}{dx}}{\frac{dz}{dx}} = \frac{x^{\sin x} \left(\frac{\sin x}{x} + \cos x \log x \right)}{(\sin x)^x (x \cot x + \log \sin x)}. \]

Answer 2

To find the rate of change of \(x^{\sin x}\) with respect to \((\sin x)^x\), we need to calculate the derivative of \(x^{\sin x}\) with respect to \(x\) and the derivative of \((\sin x)^x\) with respect to \(x\), and then divide these two derivatives.

Step 1: Differentiate \(x^{\sin x}\) with respect to \(x\).

Let \(y = x^{\sin x}\). Taking the natural logarithm on both sides, we get:

\(\ln y = \sin x \ln x\)

Differentiate both sides with respect to \(x\):

\(\frac{1}{y}\frac{dy}{dx} = \cos x \ln x + \frac{\sin x}{x}\)

\(\frac{dy}{dx} = x^{\sin x}\left(\cos x \ln x + \frac{\sin x}{x}\right)\)

Step 2: Differentiate \((\sin x)^x\) with respect to \(x\).

Let \(z = (\sin x)^x\). Taking the natural logarithm on both sides, we get:

\(\ln z = x \ln (\sin x)\)

Differentiate both sides with respect to \(x\):

\(\frac{1}{z}\frac{dz}{dx} = \ln (\sin x) + x \frac{\cos x}{\sin x}\)

\(\frac{1}{z}\frac{dz}{dx} = \ln (\sin x) + x \cot x\)

\(\frac{dz}{dx} = (\sin x)^x (\ln (\sin x) + x \cot x)\)

Step 3: Find the rate of change.

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

\(\frac{\frac{dy}{dx}}{\frac{dz}{dx}} = \frac{x^{\sin x} \left(\cos x \ln x + \frac{\sin x}{x}\right)}{(\sin x)^x (\ln (\sin x) + x \cot x)}\)

The correct option is:

\(\frac{x^{\sin x} \left(\frac{\sin x}{x} + \cos x \log x \right)}{(\sin x)^x (x \cot x + \log \sin x)}\)