← Unit 3: Differentiation — Composite, Implicit & Inverse Functions
Derivatives of Inverse Functions
The derivative of an inverse function at a point relates to the reciprocal of the original function's derivative at the corresponding point; inverse trig functions have their own memorizable derivative formulas.
The inverse function derivative formula
If g is the inverse of f, then g'(x) = 1 / f'(g(x)). This comes from reflecting the tangent line slope across the line y=x — reflecting a slope m gives slope 1/m.
Key inverse trig derivatives: d/dx[arcsin(x)] = 1/√(1-x²), d/dx[arctan(x)] = 1/(1+x²). These are worth memorizing directly.
Worked Example
If f(x) = x³ + 2x and g is its inverse, find g'(3), given that f(1) = 3.
- Since f(1)=3, we know g(3)=1 (inverse functions swap input/output).
- f'(x) = 3x² + 2, so f'(1) = 3(1)+2 = 5.
- g'(3) = 1/f'(g(3)) = 1/f'(1) = 1/5.
Answer: g'(3) = 1/5
Key terms (2)
- Inverse function derivative
- g'(x) = 1/f'(g(x)) when g is the inverse of f.
- d/dx[arctan x]
- 1/(1+x²)
Practice Quiz
Question 1 of 3Score so far: 0/0
d/dx[arctan(x)] = ?