Question

Differentiate \[{10}^{ \log \sin x }\] ?

Answer 1

\[\text{ Let y} = {10}^{\log \sin x} . . . \left( i \right)\]
\[\text{ Taking log on both sides }, \]
\[\log y = \log {10}^{\log \sin x} \]
\[ \Rightarrow \log y = \log \sin x \log10 \]
\[\text{ Differentiating with respect to x}, \]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \log10\frac{d}{dx}\log \sin x \]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \log10\frac{1}{\sin x}\frac{d}{dx}\left( \sin x \right)\]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \log10\left( \frac{1}{\sin x} \right)\left( \cos x \right)\]
\[ \Rightarrow \frac{dy}{dx} = y\left[ \log10 \times \cot x \right]\]
\[ \Rightarrow \frac{dy}{dx} = {10}^{\log \sin x} \times \log10 \times \cot x \left[ \text{ using equation } \left( i \right) \right]\]