COMEDK 2025 Afternoon Solved Paper

Concept:We use the formula for derivative of a logarithmic function and the derivative of an exponential, then apply the chain rule for differentiating one function with respect to another.Explanation:Let $y = \log_a x = \frac{\ln x}{\ln a}$.Let $u = a^x = e^{x \ln a}$.We need $\frac{dy}{du}$. By the chain rule, $\frac{dy}{du} = \frac{dy/dx}{du/dx}$.Compute $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{1}{x \ln a}$.Compute $\frac{du}{dx}$: $\frac{du}{dx} = a^x \ln a$.Now divide: $\frac{dy}{du} = \frac{\frac{1}{x \ln a}}{a^x \ln a} = \frac{1}{x a^x (\ln a)^2}$.Since $\ln a$ can be written as $\log a$ (natural log), the result matches the form $\frac{1}{x a^x (\log a)^2}$.Answer:$\frac{1}{x a^x (\log a)^2}$ corresponds to option B.