Question

Given that \( y = (\sin x)^x \cdot x^{\sin x} + a^x \), find \( \frac{dy}{dx} \).

Hint:

For functions involving powers and products, logarithmic differentiation simplifies the process.

Answer 1

Step 1: Express \( y \) in terms of simpler functions
Thus: \[ \frac{dy}{dx} = \frac{du}{dx} + \frac{d}{dx} (a^x). \] 
Step 2: Differentiate \( u \)
Taking logarithm: \[ \log u = x \log(\sin x) + \sin x \log x. \] Differentiating with respect to \( x \): \[ \frac{1}{u} \frac{du}{dx} = \log(\sin x) + x \cot x + \log x \cdot \cos x + \sin x \cdot \frac{1}{x}. \] Thus: \[ \frac{du}{dx} = u \cdot [\log(\sin x) + x \cot x + \sin x \log x + \cos x \log x]. \] 
Step 3: Add the derivative of \( a^x \)
The derivative of \( a^x \) is: \[ \frac{d}{dx} (a^x) = a^x \log a. \] 
Conclusion: The derivative is: \[ \frac{dy}{dx} = (\sin x)^x x^\sin x [\log(\sin x) + x \cot x + \sin x \log x + \cos x \log x] + a^x \log a. \]