Which expression equals for a curve defined parametrically by differentiable…

MATHEMATICS · Single correct · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

Which expression equals

\dfrac{d^{2}y}{dx^{2}}

for a curve defined parametrically by differentiable functions

x=f(t)

and

y=g(t)

with

\dfrac{dx}{dt}\neq 0

? (a)

\displaystyle \frac{\dfrac{d^{2}y}{dt^{2}}}{\left(\dfrac{dx}{dt}\right)^{2}}

(b)

\displaystyle \frac{\dfrac{d}{dt}\!\left(\dfrac{dy}{dx}\right)}{\dfrac{dx}{dt}}

(c)

\displaystyle \frac{d}{dt}\!\left(\dfrac{dy}{dx}\right)\,\left(\dfrac{dx}{dt}\right)

(d)

\displaystyle \frac{d}{dt}\!\left(\dfrac{dy}{dx}\right)