If where and are twice differentiable functions

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

Problem

If y=f(g(x))y = f(g(x)) where ff and gg are twice differentiable functions, which of the following equals d2ydx2\dfrac{d^{2}y}{dx^{2}}? (a) f′′(g(x)) g′′(x)+f′(g(x)) [g′(x)]2f''(g(x))\,g''(x)+f'(g(x))\,[g'(x)]^{2} (b) f′′(g(x)) [g′(x)]2+f′(g(x)) g′′(x)f''(g(x))\,[g'(x)]^{2}+f'(g(x))\,g''(x) (c) f′′(g(x)) [g′(x)]2−f′(g(x)) g′′(x)f''(g(x))\,[g'(x)]^{2}-f'(g(x))\,g''(x) (d) 2 f′′(g(x)) [g′(x)]2+f′(g(x)) g′′(x)2\,f''(g(x))\,[g'(x)]^{2}+f'(g(x))\,g''(x)

Related theory

Browse the problem library