The derivative of an inverse function tells how fast the inverse changes compared with the original function. If a function f sends x to y, then its inverse f^-1 sends y back to x. This matters because many real relationships are easier to measure in one direction but needed in the reverse direction.
The key idea is that inverse functions swap the roles of input and output.
Key Facts
- If f(a) = b, then f^-1(b) = a.
- Derivative of an inverse: (f^-1)'(b) = 1 / f'(a), where b = f(a).
- Equivalent formula: (f^-1)'(x) = 1 / f'(f^-1(x)).
- The graph of f^-1 is the reflection of the graph of f across the line y = x.
- Inverse slopes are reciprocals at corresponding reflected points, as long as f'(a) is not 0.
- Example: If f(x) = x^3 + 1, then f'(x) = 3x^2 and (f^-1)'(9) = 1 / f'(2) = 1 / 12 because f(2) = 9.
Vocabulary
- Inverse function
- An inverse function reverses the input and output of a function, so f^-1(f(x)) = x when both are defined.
- Derivative
- A derivative gives the instantaneous rate of change of a function at a point.
- Reciprocal slope
- A reciprocal slope is the value 1 divided by the original slope.
- Corresponding points
- Corresponding points on a function and its inverse have swapped coordinates, such as (a, b) and (b, a).
- One-to-one function
- A one-to-one function gives each output from exactly one input, which allows it to have an inverse function.
Common Mistakes to Avoid
- Using (f^-1)'(x) = 1 / f'(x), which is wrong because the derivative of f must be evaluated at f^-1(x), not usually at x.
- Forgetting to find the matching input a first, which is wrong because the formula needs the point where f(a) equals the inverse input.
- Applying the formula when f'(a) = 0, which is wrong because division by zero is undefined and the inverse may have a vertical tangent there.
- Thinking the inverse slope is always negative, which is wrong because reflection across y = x makes slopes reciprocal, not opposite in sign.
Practice Questions
- 1 Let f(x) = 2x + 5. Find (f^-1)'(11).
- 2 Let f(x) = x^3 + x + 1. Since f(1) = 3, find (f^-1)'(3).
- 3 A function has slope 4 at the point (2, 7). Explain what this tells you about the slope of its inverse at the point (7, 2), and why.