$\text { Find the second derivative } \frac { d ^ { 2 } y } { d x ^ { 2 } } \text { of the parametric equations } x = 4 t , y = 8 t - 3 \text {. }$ Round your answer to two decimal places, if necessary.
A) $0$
B) $2.00$
C) $- \infty$
D) $0.50$
E) $\infty$

Question

$\text { Find the second derivative } \frac { d ^ { 2 } y } { d x ^ { 2 } } \text { of the parametric equations } x = 4 t , y = 8 t - 3 \text {. }$ Round your answer to two decimal places, if necessary. 
A) $0$
B) $2.00$
C) $- \infty$
D) $0.50$
E) $\infty$

Quizplus · Accepted Answer

The second derivative of a function in parametric form, $y$ as a function of $x$ through a parameter $t$, is given by $\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}$. For the given equations $x = 4t$ and $y = 8t - 3$, we first find $\frac{dx}{dt} = 4$ and $\frac{dy}{dt} = 8$. Then, $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{8}{4} = 2$, which is constant. Since $\frac{dy}{dx}$ is constant, its derivative with respect to $t$ (or $x$) is $0$, indicating that the second derivative $\frac{d^2y}{dx^2}$ is $0$.