3.3 Differentiating Inverse Functions

Reciprocal slopes

If g = f⁻¹, then (f⁻¹)'(x) = 1 / f'(f⁻¹(x)). The slopes of a function and its inverse are reciprocals at corresponding points.

Using a point

If f(a) = b (so f⁻¹(b) = a), then (f⁻¹)'(b) = 1/f'(a). Example: f(2) = 5 and f'(2) = 4 give (f⁻¹)'(5) = 1/4.

Key terms

Inverse derivative

(f⁻¹)'(x) = 1 / f'(f⁻¹(x)).

Reciprocal slopes

A function and its inverse have reciprocal slopes.

Corresponding points

f(a) = b means f⁻¹(b) = a.

Point formula

(f⁻¹)'(b) = 1/f'(a).

Zero slope

If f'(a) = 0, the inverse has an undefined (vertical) slope.

Try an AP-style question

f(2) = 5 and f'(2) = 4. If g is the inverse of f, what is g'(5)?

1. 1/2
2. 1/4
3. 1/5
4. 4