If and are twice differentiable with

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

Problem

If x=f(t)x=f(t) and y=g(t)y=g(t) are twice differentiable with f′(t)≠0f'(t)\neq0, which of the following gives d2ydx2\dfrac{d^{2}y}{dx^{2}} in terms of tt? (a) f′′(t) g′(t)[f′(t)]3\displaystyle\frac{f''(t)\,g'(t)}{[f'(t)]^{3}} (b) g′′(t)[f′(t)]2\displaystyle\frac{g''(t)}{[f'(t)]^{2}} (c) f′(t) g′′(t)−g′(t) f′′(t)[f′(t)]3\displaystyle\frac{f'(t)\,g''(t)-g'(t)\,f''(t)}{[f'(t)]^{3}} (d) f′′(t) g′(t)−g′′(t) f′(t)[f′(t)]3\displaystyle\frac{f''(t)\,g'(t)-g''(t)\,f'(t)}{[f'(t)]^{3}}

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