MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
Consider three differentiable functions f, g, and h with composite relationship y = f(g(h(x))). Assume that all relevant derivatives exist on an open interval containing x = a.
(a) Describe—in complete sentences and without performing any calculations—the sequential dependence of the instantaneous rate of change of y with respect to x at x = a on the individual derivatives of f, g, and h.
(b) Write an expression, using derivative notation, that captures the general structure you described in part (a). Your answer should make clear which derivative is evaluated at which function value.
(c) Explain why omitting any one of the intermediate derivatives identified in part (b) would lead to an incorrect numerical value for dxdy at x = a. Provide a specific hypothetical consequence to illustrate your explanation.