For differentiable functions and with

MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

For differentiable functions x(t)x(t) and y(t)y(t) with dxdt≠0\dfrac{dx}{dt}\neq 0, which of the following expressions is equal to d2ydx2\dfrac{d^{2}y}{dx^{2}}? (a) (dydt)(d2xdt2)(dxdt)3\displaystyle \frac{\left(\dfrac{dy}{dt}\right)\left(\dfrac{d^{2}x}{dt^{2}}\right)}{\left(\dfrac{dx}{dt}\right)^{3}} (b) ddt ⁣(dydx)dxdt\displaystyle \frac{\dfrac{d}{dt}\!\left(\dfrac{dy}{dx}\right)}{\dfrac{dx}{dt}} (c) d2ydt2(dxdt)2\displaystyle \frac{\dfrac{d^{2}y}{dt^{2}}}{\left(\dfrac{dx}{dt}\right)^{2}} (d) ddt ⁣(dydx)d2xdt2\displaystyle \frac{\dfrac{d}{dt}\!\left(\dfrac{dy}{dx}\right)}{\dfrac{d^{2}x}{dt^{2}}}

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