If y=e^nx, then d^2y dx^2. d^2x dy^2 is equal to :

$y={e}^{nx}\Rightarrow \frac{dy}{dx}=n{e}^{nx}$y=enx⇒dxdy​=nenx ${\because \frac{d}{dx}{e}^{ax}=a{e}^{ax}}$∵dxd​eax=aeax

$$\Rightarrow \frac{{d}^{2}y}{d{x}^{2}}={n}^{2}{e}^{nx}$$⇒dx2d2y​=n2enx

$$\frac{dy}{dx}=n{e}^{nx}\Rightarrow \frac{dx}{dy}=\frac{1}{n{e}^{nx}}=\frac{1}{n}{e}^{-nx}$$dxdy​=nenx⇒dydx​=nenx1​=n1​e−nx

$$\frac{{d}^{2}x}{d{y}^{2}}=\frac{d}{dy}(\frac{dx}{dy})$$dy2d2x​=dyd​(dydx​)

$$=\frac{d}{dx}(\frac{dx}{dy})\cdot \frac{dx}{dy}$$=dxd​(dydx​)⋅dydx​

$$=\frac{d}{dx}(\frac{1}{n}{e}^{-nx})\cdot \frac{{e}^{-nx}}{n}$$=dxd​(n1​e−nx)⋅ne−nx​

$$=\frac{1}{n}(-n){e}^{-nx}\cdot \frac{{e}^{-nx}}{n}$$=n1​(−n)e−nx⋅ne−nx​

$$=-\frac{1}{n}{e}^{-2nx}$$=−n1​e−2nx

$$\Rightarrow \frac{{d}^{2}y}{d{x}^{2}}.\frac{{d}^{2}x}{d{y}^{2}}={n}^{2}{e}^{nx}\times (\frac{-{e}^{-2nx}}{n})$$⇒dx2d2y​.dy2d2x​=n2enx×(n−e−2nx​)

$$=-n{e}^{-nx}$$=−ne−nx