The inverse function theorem explains how the derivative of a function is related to the derivative of its inverse. It matters because inverse functions appear throughout calculus, including logarithms, inverse trigonometric functions, and solving equations by reversing a process. The central idea is that if a function has a nonzero slope at a point and is locally invertible there, then its inverse is also differentiable at the matching point.
On a graph, a function and its inverse are mirror images across the line y = x.
Key Facts
- If f is differentiable near a, f'(a) ≠ 0, and f has an inverse near a, then (f^-1)'(f(a)) = 1 / f'(a).
- If b = f(a), then (f^-1)'(b) = 1 / f'(a).
- The slope of the inverse at (f(a), a) is the reciprocal of the slope of f at (a, f(a)).
- A nonzero derivative f'(a) means the tangent line is not horizontal, so the reflected inverse tangent is not vertical.
- The graphs of y = f(x) and y = f^-1(x) are reflections across y = x.
- To find (f^-1)'(b), solve f(a) = b first, then compute 1 / f'(a).
Vocabulary
- Inverse function
- An inverse function reverses the input and output of a function, so f^-1(f(x)) = x for allowed values of x.
- Local inverse
- A local inverse is an inverse that exists only on a small interval near a chosen point.
- Derivative
- The derivative gives the instantaneous rate of change or slope of a function at a point.
- Reciprocal slope
- A reciprocal slope is formed by switching numerator and denominator, such as changing m into 1/m.
- Reflection across y = x
- Reflection across y = x swaps every point (x, y) on a graph with the point (y, x).
Common Mistakes to Avoid
- Using (f^-1)'(a) = 1 / f'(a) directly is wrong because the input to the inverse derivative must be an output value of f. If b = f(a), then the correct formula is (f^-1)'(b) = 1 / f'(a).
- Forgetting to solve f(a) = b first is wrong because the derivative f'(a) must be evaluated at the original x-value, not at the inverse input b.
- Applying the theorem when f'(a) = 0 is wrong because the reciprocal 1 / f'(a) is undefined and the inverse may fail to be differentiable there.
- Assuming a global inverse always exists is wrong because a function may only be one-to-one on a restricted interval. The theorem only needs an inverse near the point.
Practice Questions
- 1 Let f(x) = x^3 + x. Find (f^-1)'(2).
- 2 Let f(x) = e^x + 2x. Find (f^-1)'(1).
- 3 A differentiable function has f(3) = 7 and f'(3) = -4. Explain what point lies on the graph of f^-1 and what the slope of f^-1 is at that point.