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.
Understanding Calculus: The Derivative of an Inverse Function
The reciprocal rule comes from the chain rule. Think of starting with an input, applying a function, then immediately applying its inverse. You return to the same input, so the overall rate of change is one.
The chain rule says that this overall rate is the slope of the inverse, measured at the output value, multiplied by the slope of the original function, measured at the matching input value. For their product to be one, each nonzero slope must be the reciprocal of the other.
This reasoning explains why it is important to match the two points correctly. The inverse slope is not usually found by using the same number in the original derivative.
A reciprocal can behave very differently from the original number. A steep positive slope such as twelve becomes a small positive inverse slope of one twelfth. This means the inverse changes only a little when its input changes.
A shallow original slope such as one fifth becomes an inverse slope of five, so the inverse changes quickly. If the original slope is zero, there is no ordinary reciprocal. On the graph, a horizontal tangent of the original becomes a vertical tangent of the inverse.
At that point, the inverse derivative is undefined. This is not a small technical detail. It tells you that a tiny change in the inverse input can cause a large change in its output.
An inverse must be a true function before this rule can be used across a whole interval. The original function needs to give each output only once on that interval. For example, a parabola fails this test if its full graph is used, since two different inputs can have the same output.
Restricting the parabola to one side of its turning point creates an interval where an inverse can exist. Students often meet this issue with square roots, inverse trigonometric functions, and logarithms. The restrictions are chosen so that each allowed output leads back to one clear input.
Inverse derivatives appear whenever a measured quantity must be converted back into the quantity that produced it. A logarithm reverses an exponential relationship, so its derivative describes how changes in a logarithmic scale connect to changes in the original scale. Calibration curves work similarly.
A device may report a voltage, while a scientist needs the temperature that produced that voltage. When solving problems, first identify the input value of the original function that creates the given inverse input. Then find the original slope at that input and take its reciprocal.
Keep track of units as well. If the original slope is output units per input unit, the inverse slope has input units per output unit. This unit reversal is a useful check on the result.
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.