Question

y=f(x) be a real valued twice differentiable function defined on R, then
$\frac{d^{2}y}{d{x}^2}{\left(\frac{dx}{dy}\right)}^3+\frac{d^{2}x}{d{y}^2}$=

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

$\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)=\left(\frac{d}{\frac{dx}{dy}}\right)=\frac{d}{dy}\left(\frac{1}{\frac{dx}{dy}}\right)\frac{dy}{dx}=\frac{\frac{d^{2}x}{d{y}^2}}{{\left(\frac{dx}{dy}\right)}^2}\frac{dy}{dx}=-\frac{\frac{d^{2}x}{d{y}^2}}{{\left(\frac{dx}{dy}\right)}^3}\Rightarrow \frac{d^{2}y}{d{x}^2}{\left(\frac{dx}{dy}\right)}^3+\frac{d^{2}x}{d{y}^2}=0$