Question

If $x=4t^2+5,y=6t^2+7t+3$, find $\frac{d^{2}y}{d{x}^2}$.

Answer 1

Differentiate $x=4t^2+5$ with respect to $t$,

$\frac{dx}{dt}=8t$

Again differentiate with respect to $t$,

$\frac{d^{2}x}{d{t}^2}=8$ (1)

Differentiate $y=6t^2+7t+3$ with respect to $t$,

$\frac{dy}{dt}=12t+7$

Again differentiate with respect to $t$,

$\frac{d^{2}y}{d{t}^2}=12$ (2)

Divide equation (2) by equation (1),

$\frac{d^{2}y}{d{x}^2}=\frac{12}{8}$

$\frac{d^{2}y}{d{x}^2}=\frac{3}{2}$