Differentiating Inverse Functions
Find the corresponding original input before taking a nonzero inverse slope reciprocal.
College Board Topic 3.3 · CED p. 72 (PDF p. 77) · CED effective Fall 2020; project course outline 2026–27.
An inverse swaps inputs and outputs: if f(a)=b, then f⁻¹(b)=a. The graphs reflect across y=x. For a differentiable inverse g=f⁻¹, the identity f(g(x))=x gives f′(g(x))g′(x)=1, so g′(b)=1/f′(a), with a=g(b) and f′(a)≠0.
In familiar smooth one-to-one models, a nonzero local slope gives the inverse a reciprocal slope. State the inverse or one-to-one interval, identify the original input corresponding to the requested inverse input, and check that the original derivative is nonzero. The original and inverse rates have reciprocal units.
If f(2)=7 and f′(2)=4, then (f⁻¹)′(7)=1/4. It is not 1/f′(7). A zero original slope prevents use of a finite reciprocal derivative. For x³, the inverse x^(1/3) has a vertical tangent at 0 and no finite derivative there. “Inverse” also does not mean reciprocal: f⁻¹(x) and 1/f(x) are different functions.
Let's work through it
- Suppose f is one-to-one and its differentiable inverse g satisfies f(3)=8 and f′(3)=−2. Then g(8)=3. Reason: an inverse exchanges the input-output pair.
- Use g′(8)=1/f′(g(8))=1/f′(3). Reason: differentiate f(g(x))=x with the chain rule.
- The inverse slope is −1/2. Reason: the original slope is nonzero, so its reciprocal is finite.
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Question 1
If f(1)=4 and f′(1)=5, find the differentiable inverse slope at 4.
See every recall question with its answer
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Retrieval 1
If f(1)=4 and f′(1)=5, find the differentiable inverse slope at 4.
Worked answer and reason
1/5.
The corresponding original input is 1.
Retrieval 2
Is f⁻¹(x) the same as 1/f(x)?
Worked answer and reason
No.
The inverse undoes an input-output mapping; the reciprocal divides 1 by an output.
Catch the tempting mistake
- To find the inverse slope at b, use 1/f′(b).
- First find a with f(a)=b; the needed original derivative is f′(a).
Calculator workspace and limits
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Put it to work
- Unit 3 problem 14 · No calculator · numerical
- Unit 3 problem 15 · No calculator · analytic
- Unit 3 problem 16 · No calculator · analytic
- Unit 3 problem 17 · Graphing calculator required · multiple
- Unit 3 problem 18 · No calculator · analytic
For the teacher
- Use a table containing both f and f′ values at several inputs.
- Ask students to explain how reflected tangent slopes become reciprocal.
Glossary and deeper reading · 5 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Differentiability
Differentiability means that a function has a finite derivative at the point under discussion. Differentiability at a point implies continuity there, but continuity alone does not imply differentiability. For example, |x| is continuous at 0 but its left and right tangent slopes differ, so it is not differentiable there. Corners, cusps, vertical tangents, and discontinuities can prevent a finite derivative.
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One-to-one function
A function is one-to-one when different inputs always have different outputs. Equivalently, each horizontal line meets its graph at most once. For example, x³ is one-to-one on the real numbers, whereas x² is not because 2 and −2 have the same output. This property allows a function’s inverse relation to be a function.
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Interval
An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.
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Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.
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Vertical tangent
A vertical tangent is a tangent line parallel to the vertical axis, with no finite slope. A graph can have a vertical tangent even though its derivative does not exist as a finite real number at that input. For example, y=∛x has the vertical tangent x=0 at the origin as its nearby slopes tend to positive infinity. This case is separate from the finite-slope tangent approximation available at a differentiable point.