Mathematics - Differentiation Question with Solution | TestHub

$y=e^{nx} \Rightarrow \frac{dy}{dx}=n{e}^{nx}$$\left\{\because \frac{d}{dx}{e}^{ax}=a{e}^{ax}\right\}$

$\Rightarrow \frac{d^{2}y}{d{x}^2}=n^2{e}^{nx}$

$\frac{dy}{dx}=n{e}^{nx}\Rightarrow \frac{dx}{dy}=\frac{1}{n{e}^{nx}}=\frac{1}{n}{e}^{-nx}$

$\frac{d^{2}x}{d{y}^2}=\frac{d}{dy}\left(\frac{dx}{dy}\right)$

$=\frac{d}{dx}\left(\frac{dx}{dy}\right)·\frac{dx}{dy}$

$=\frac{d}{dx}\left(\frac{1}{n}{e}^{-nx}\right)·\frac{e^{-nx}}{n}$

$=\frac{1}{n}\left(-n\right)e^{-nx}·\frac{e^{-nx}}{n}$

$=-\frac{1}{n}{e}^{-2nx}$

$\Rightarrow \frac{d^{2}y}{d{x}^2}.\frac{d^{2}x}{d{y}^2}=n^2{e}^{nx}\times \left(\frac{-e^{-2nx}}{n}\right)$

$=-n{e}^{-nx}$