AP® Calculus AB · Unit 3 · Topic 3.3
Differentiating Inverse Functions
The derivative of an inverse function at a point equals one over the derivative of the original function at the corresponding input: (f−1)'(y) = 1/f'(x), where y = f(x).
Swapping x and y to build an inverse also swaps rise and run. If f has slope f'(x) = rise/run at a point, then f−1, which reflects that point over the line y = x, has slope run/rise there — the reciprocal. That is the whole idea: (f−1)'(y) = 1/f'(x), as long as you evaluate f' at the right input. The tricky part is bookkeeping. If f(x) = y, then (f−1)'(y) is 1 over f'(x) — not 1 over f' evaluated at y. You never need an actual formula for f−1; you just need a point (x, y) on f and the slope f' at that x.
Work it through for f(x) = x3 + 2x + 1. The point where y = 4 means solving x3 + 2x + 1 = 4, which gives x = 1. Then f'(x) = 3x2 + 2, so f'(1) = 5. The inverse's derivative at y = 4 is 1/5. No need to ever find f−1(x) explicitly — that is the entire point of this rule.
Where f has a horizontal tangent, f'(x) = 0, and the reciprocal formula breaks: 1/0 is undefined. Geometrically the reflection turns that flat tangent into a vertical one on f−1, so the inverse simply isn't differentiable there. This same reciprocal-slope idea is what generates the derivatives of arcsine, arctangent, and the other inverse trig functions next: each comes from applying this formula to sine, tangent, and friends, then rewriting the denominator in terms of x using a right triangle or a Pythagorean identity.
Key terms
- Inverse function
- A function that reverses another, sending each output y back to its original input x.
- Reciprocal slope
- The flipped value of a slope, run over rise instead of rise over run.
- (f−1)'(y) = 1/f'(x)
- The rule giving the inverse's derivative at y in terms of the original function's derivative at the matching x.
- Horizontal tangent
- A point where f'(x) = 0, which corresponds to a vertical, undefined tangent on the inverse.
Watch it explained
- How it worksHow does reflecting a curve over the line y = x swap rise and run?
- Worked exampleFind the derivative of the inverse of f(x) = x3 + 2x + 1 at the point where y = 4.
- Common mistakeWhy isn't the inverse's derivative one over the original derivative at the same x?
- Going furtherWhat happens to the inverse's derivative where the original has a horizontal tangent?
- Ties togetherHow does this general inverse-function derivative formula produce the arctangent and arcsine derivatives coming up next?
Check yourself
End of topic.