Question

Given the parametric equations $x=f\left(t\right),y=g\left(t\right)$, then $\frac{d^{2}y}{d{x}^2}$ equals

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

Answer 1

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

We have $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$
$\therefore \frac{d^{2}y}{d{x}^2}=\frac{d}{dt}\left(\frac{\frac{dy}{dt}}{\frac{dx}{dt}}\right)\frac{dt}{dx}$
$=\frac{\frac{d^{2}y}{d{t}^2}\left(\frac{dx}{dt}\right)-\left(\frac{d^{2}x}{d{t}^2}\right)\frac{dy}{dt}}{{\left(\frac{dx}{dt}\right)}^2}\times \frac{1}{\frac{dx}{dt}}$
$=\frac{\left(\frac{dx}{dt}\right).\left(\frac{d^{2}y}{d{t}^2}\right)-\left(\frac{d^{2}x}{d{t}^2}\right).\left(\frac{dy}{dt}\right)}{{\left(\frac{dx}{dt}\right)}^3}$.