3.3 Differentiating Inverse Functions

Derivative of an Inverse Function

If f is differentiable and invertible near a point, the derivative of its inverse is given by (f⁻¹)'(x) = 1 / f'(f⁻¹(x)). This comes from differentiating both sides of f(f⁻¹(x)) = x using the chain rule. The key idea: reciprocal slopes correspond to reflected graphs over y = x.

Finding Values Without a Formula

You often don't need an explicit formula for f⁻¹. If you know f(a) = b and f'(a), you can find (f⁻¹)'(b) using (f⁻¹)'(b) = 1 / f'(a). This is common on the exam: use given table values or known points to compute the derivative of the inverse at a specific point.

Why It Works Graphically

Since f and f⁻¹ are reflections across the line y = x, their slopes at corresponding points are reciprocals. If f has slope m at (a, b), then f⁻¹ has slope 1/m at (b, a). This reciprocal relationship is the geometric heart of the inverse derivative formula.

Key terms

Inverse derivative formula

(f⁻¹)'(x) = 1 / f'(f⁻¹(x))

How to derive the inverse derivative formula

Differentiate f(f⁻¹(x)) = x using the chain rule: f'(f⁻¹(x)) · (f⁻¹)'(x) = 1.

If f(a) = b, then (f⁻¹)'(b) = ?

1 / f'(a) — the reciprocal of f's derivative at the corresponding input.

Geometric meaning of inverse derivatives

Since f and f⁻¹ are reflections over y = x, their slopes at corresponding points are reciprocals.

What happens if f'(a) = 0?

(f⁻¹)'(b) is undefined, since it would require dividing by zero.

Try an AP-style question

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

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