Question

$\frac{d}{dx}\left(\frac{dx}{dy}\right)\cdot \frac{d}{dy}\left(\frac{dy}{dx}\right)$ is equal to

• A. ${\left(\frac{d^{2}y}{d{x}^2}\right)}^3{\left(\frac{dy}{dx}\right)}^{-2}$
• B. $-{\left(\frac{d^{2}y}{d{x}^2}\right)}^2{\left(\frac{dy}{dx}\right)}^{-3}$
• C. $-{\left(\frac{d^{2}y}{d{x}^2}\right)}^3{\left(\frac{dy}{dx}\right)}^{-3}$
• D. ${\left(\frac{d^{2}y}{d{x}^2}\right)}^2{\left(\frac{dy}{dx}\right)}^{-3}$

Answer 1

The correct option is B $-{\left(\frac{d^{2}y}{d{x}^2}\right)}^2{\left(\frac{dy}{dx}\right)}^{-3}$

$\text{Let,}y=f\left(x\right)\Rightarrow \frac{dy}{dx}=f^{\prime }\left(x\right)\Rightarrow \frac{dx}{dy}=\frac{1}{f^{\prime }\left(x\right)}\therefore \frac{d}{dx}\left(\frac{dx}{dy}\right)=-\frac{1}{\left[f^{\prime }\right(x){]}^2}\cdot f^{\prime \prime }\left(x\right)=-\frac{\frac{d^{2}y}{d{x}^2}}{{\left(\frac{dy}{dx}\right)}^2}$
$\text{Now,}\frac{d}{dy}\left(\frac{dy}{dx}\right)=\frac{d}{dx}\left(\frac{dy}{dx}\right)\cdot \frac{dx}{dy}=\frac{d^{2}y}{d{x}^2}\cdot \frac{1}{\frac{dy}{dx}}$
$\therefore \frac{d}{dx}\left(\frac{dx}{dy}\right)\cdot \frac{d}{dy}\left(\frac{dy}{dx}\right)=\frac{-{\left(\frac{d^{2}y}{d{x}^2}\right)}^2}{{\left(\frac{dy}{dx}\right)}^3}=-{\left(\frac{d^{2}y}{d{x}^2}\right)}^2{\left(\frac{dy}{dx}\right)}^{-3}$