Let be a differentiable one-to-one function with inverse
MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
Let f be a differentiable one-to-one function with inverse g. Which of the following best justifies that the slopes of the graphs of f and g at corresponding points are reciprocals?
(a) Because differentiating f(g(x))=x gives f′(g(x))=g′(x)1, it follows that g′(x)=f′(g(x)), so the slopes are equal at corresponding points.
(b) Because the equation f(g(x))=x differentiates to f′(g(x))g′(x)=1, the chain rule forces g′(x)=f′(g(x))1.
(c) Because f and g are reflections across the x-axis, their slopes differ by a factor of −1.},{
(d) Because the Mean Value Theorem guarantees some c with f′(c)=g′(c), the slopes must therefore be reciprocals for all c.