Mathwords logoMathwords

Derivative of an Inverse Function — Definition, Formula & Examples

The derivative of an inverse function gives you the slope of f⁻¹ at a point by flipping the derivative of the original function. Specifically, you evaluate 1 divided by f′ at the corresponding point on f.

If f is a differentiable function with a differentiable inverse f⁻¹, and if f'(f⁻¹(a)) ≠ 0, then the derivative of the inverse function at a is given by (f−1)′(a)=1f′ ⁣(f−1(a))(f^{-1})'(a) = \dfrac{1}{f'\!\bigl(f^{-1}(a)\bigr)}.

Key Formula

(f−1)′(a)=1f′ ⁣(f−1(a))(f^{-1})'(a) = \frac{1}{f'\!\bigl(f^{-1}(a)\bigr)}
Where:
  • ff = The original differentiable function
  • f−1f^{-1} = The inverse of f
  • aa = The input value at which you want the derivative of the inverse
  • f′f' = The derivative of the original function f

How It Works

To use this formula, you need three things: the point aa where you want the derivative of f−1f^{-1}, the value f−1(a)f^{-1}(a) (the x-value on the original function that maps to aa), and the derivative f′f' evaluated at that x-value. You then take the reciprocal of f′f' at that point. This works because inverse functions swap inputs and outputs, so slopes get flipped. The formula applies to any invertible, differentiable function — not just inverse trig functions, which are a special case with their own memorized derivative formulas.

Worked Example

Problem: Let f(x) = x³ + x. Given that f(2) = 10, find (f⁻¹)'(10).
Identify f⁻¹(a): Since f(2) = 10, the inverse function satisfies f⁻¹(10) = 2.
f−1(10)=2f^{-1}(10) = 2
Find f′(x) and evaluate at f⁻¹(a): Differentiate f(x) = x³ + x to get f′(x) = 3x² + 1. Then evaluate at x = 2.
f′(2)=3(2)2+1=13f'(2) = 3(2)^2 + 1 = 13
Apply the formula: Take the reciprocal of f′ evaluated at that point.
(f−1)′(10)=1f′(2)=113(f^{-1})'(10) = \frac{1}{f'(2)} = \frac{1}{13}
Answer: (f−1)′(10)=113(f^{-1})'(10) = \dfrac{1}{13}

Why It Matters

This formula appears on the AP Calculus AB and BC exams, often in problems where you cannot explicitly solve for f⁻¹. It also provides the theoretical foundation for deriving the specific formulas for derivatives of inverse trig functions like arcsin, arctan, and arcsec.

Common Mistakes

Mistake: Evaluating f′ at a instead of at f⁻¹(a)
Correction: You must first find the x-value where f(x) = a, then plug that x-value into f′. The derivative is 1/f′(f⁻¹(a)), not 1/f′(a).

Related Terms