Question

If $f\text{'}\left(1\right)=2\mathrm{and}y=f\left({\log }_{e}x\right),\mathrm{find}\frac{dy}{dx}atx=e$.

Answer 1

$\mathrm{We}\mathrm{have},f\text{'}\left(1\right)=2\mathrm{and}y=f\left({\log }_{e}x\right)$
Differentiate it with respect to x,
$$\begin{gathered} \frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}=f\text{'}\left({\log }_{e}x\right)\times \frac{d}{dx}\left({\log }_{e}x\right) \\ \Rightarrow \frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}=f\text{'}\left({\log }_{e}x\right)\left(\frac{1}{x}\right) \\ \Rightarrow \frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}=f\text{'}\left({\log }_{e}e\right)\left(\frac{1}{e}\right)\left[\because x=e\right] \\ \Rightarrow \frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}=f\text{'}\left(1\right)\left(\frac{1}{e}\right)\left[\because {\log }_{e}e=1\right] \\ \Rightarrow \frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}=\frac{2}{e}\left[\because f\text{'}\left(1\right)=2\right] \end{gathered}$$