MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
Let f be a differentiable function with an inverse f−1, and suppose f(a)=b with f′(a)=0.
(a) Derive an expression for (f−1)′(b) in terms of f′(a). Clearly indicate every differentiation step and any theorem or definition you invoke.
(b) The function g(x)=x3+2x+1 satisfies g(2)=15. Determine (g−1)′(15), showing all work.
(c) Interpret the numerical value found in part (b) as the slope of a tangent line on an appropriate graph, and explain what this tells you about the relative rates of change of g and g−1 at the corresponding points.