Question

If \[f'\left( 1 \right) = 2 \text { and y } = f \left( \log_e x \right), \text { find} \frac{dy}{dx} \text { at }x = e\] ?

Answer 1

\[\text { We have,}f'\left( 1 \right) = 2 \text { and y } = f\left( \log_e x \right)\]

Differentiate it with respect to x,

\[\frac{d y}{d x} = f'\left( \log_e x \right) \times \frac{d}{dx}\left( \log_e x \right)\]
\[ \Rightarrow \frac{d y}{d x} = f'\left( \log_e x \right)\left( \frac{1}{x} \right)\]
\[ \Rightarrow \frac{d y}{d x} = f'\left( \log_e e \right)\left( \frac{1}{e} \right) \left[ \because x = e \right]\]
\[ \Rightarrow \frac{d y}{d x} = f'\left( 1 \right)\left( \frac{1}{e} \right) \left[ \because \log_e e = 1 \right]\]
\[ \Rightarrow \frac{d y}{d x} = \frac{2}{e} \left[ \because f'\left( 1 \right) = 2 \right]\]