Derivatives

Derivatives of inverse functions

An inverse reverses inputs and outputs on a one-to-one branch. Its slope is reciprocal when the original derivative is nonzero.

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1. Cube and cube root

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Example graph

f(x)=x^3,\quad g(x)=\sqrt[3]{x},\quad g\prime(a^3)=\frac1{3a^2}

The inverse of x³ is the cube root. At a=1, the cubic slope is 3 and its inverse slope at y=1 is 1/3. At zero the cubic derivative vanishes, so the reciprocal rule cannot give a finite inverse slope.

f′(a): 3
g′(a³): 0.333333

The formulas identify the solid and dashed curves. Compare their values using the slider; the accompanying explanation states their mathematical relationship.

2. Exponential and logarithm

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Example graph

f(x)=e^x,\quad g(x)=\ln x,\quad g\prime(y)=\frac1y

Since ln is the inverse of exp, its derivative at y=e^a is 1/e^a=1/y. The inverse is only defined for positive y.

y=eᵃ: 1
f′(a): 1
g′(y): 1

The formulas identify the solid and dashed curves. Compare their values using the slider; the accompanying explanation states their mathematical relationship.

3. Choose an inverse branch

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Example graph

f(x)=\arcsin x,\quad f\prime(x)=\frac1{\sqrt{1-x^2}}

Sine is one-to-one on [−π/2,π/2]. Its inverse arcsin has derivative 1/√(1−x²) for −1<x<1. The slope grows near the endpoints; the formula is not finite there.

arcsin(a): 0
f′(a): 1

The solid curve is f; the dashed line is its tangent at a. Read f(a) as height and f′(a) as slope. Questions use the input stated in their text.