KCET 2025 Math Paper

To find the derivative of $\sin x$ with respect to $\log x$, we need to use the concept of implicit differentiation and the chain rule.Let $f(x)=\sin x$.We want to find $\frac{d(\sin x)}{d(\log x)}$. Using the chain rule, we have:$\frac{d(\sin x)}{d(\log x)} = \frac{d(\sin x)}{dx} \cdot \frac{dx}{d(\log x)}$ First, we calculate each part separately:$\frac{d(\sin x)}{dx} = \cos x$To find $\frac{dx}{d(\log x)}$, consider that $\frac{d(\log x)}{dx} = \frac{1}{x}$, hence:$\frac{dx}{d(\log x)} = x$ Putting it all together, we get:$\frac{d(\sin x)}{d(\log x)} = \cos x \cdot x = x \cos x$Therefore, the derivative of $\sin x$ with respect to $\log x$ is $x \cos x$.