Skip to main content
United StatesCalculusSyllabus dot point

If you know how fast a function changes, how fast does its inverse change, without ever finding a formula for the inverse?

Topic 3.3 Differentiating Inverse Functions: find the derivative of the inverse of a function at a point using the reciprocal relationship.

A focused answer to AP Calculus AB Topic 3.3, deriving and applying the inverse-function derivative formula, which relates the slope of an inverse function to the reciprocal of the original function's slope, with worked point evaluations.

Generated by Claude Opus 4.89 min answer

Reviewed by: AI editorial process; not yet individually human-reviewed

Have a quick question? Jump to the Q&A page

Jump to a section
  1. What this topic is asking
  2. The rule
  3. Why the slope inverts
  4. A reliable procedure
  5. Why you do not need a formula for the inverse
  6. Reading the conditions

What this topic is asking

The College Board (Topic 3.3) asks you to differentiate an inverse function at a point, using the fact that the slope of f−1f^{-1} at a point is the reciprocal of the slope of ff at the matching point. The power of this result is that you never need a formula for the inverse - you only need a value and a derivative of the original function.

The rule

The formula says the slope of the inverse is the reciprocal of the slope of the original, but evaluated at the corresponding point, not the same number.

Why the slope inverts

A reliable procedure

The whole method is point-matching followed by a reciprocal. Find which input of ff produces the output you care about, evaluate f′f' there, and flip it.

Why you do not need a formula for the inverse

The remarkable feature of this topic is that functions like f(x)=x5+2x+3f(x) = x^5 + 2x + 3 have no elementary inverse - there is no clean algebraic expression for g(x)g(x) - yet you can still compute g′(6)g'(6) exactly. This is possible because the inverse-derivative formula needs only two ingredients: a single matched point (found by guessing or by the original equation) and the derivative of ff, which you always have. The AP exam exploits this deliberately, choosing functions where solving for the inverse is hopeless, so that the only viable route is the reciprocal rule. The same idea underlies the derivatives of ln⁡x\ln x (the inverse of exe^x) and of the inverse trig functions in the next topic: each is found by inverting a function whose derivative you know, rather than by manipulating a formula for the inverse itself. Recognizing the pattern "I am asked for the derivative of an inverse at a point" should immediately trigger "match the point, then reciprocate f′f'".

Reading the conditions

The formula requires f′(g(x))≠0f'(g(x)) \neq 0, because you cannot divide by zero. Geometrically, if ff has a horizontal tangent (slope 00) at a point, then its reflection has a vertical tangent there, where the slope is undefined - so the inverse is not differentiable at the corresponding point. This is the one place the rule breaks down, and AP questions occasionally test awareness of it. The other requirement is that ff be one-to-one (typically because it is strictly increasing or strictly decreasing on the relevant interval), which is what guarantees the inverse exists in the first place; questions usually hand you this fact, often by noting that f′(x)>0f'(x) > 0 everywhere.

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2021 (style)1 marksSection I (multiple choice, no calculator). Let ff be differentiable with f(2)=5f(2) = 5 and f′(2)=4f'(2) = 4. If g=f−1g = f^{-1}, then g′(5)=g'(5) = (A) 44 (B) 14\frac{1}{4} (C) 15\frac{1}{5} (D) −4-4
Show worked answer →

The correct answer is (B), 14\frac{1}{4}.

The inverse-derivative formula is g′(5)=1f′(g(5))g'(5) = \frac{1}{f'(g(5))}. Since f(2)=5f(2) = 5, we have g(5)=2g(5) = 2, so g′(5)=1f′(2)=14g'(5) = \frac{1}{f'(2)} = \frac{1}{4}. The slope of the inverse at a point is the reciprocal of the slope of ff at the matching point.

AP 2022 (style)4 marksSection II (free response, no calculator). Let f(x)=x3+x+1f(x) = x^3 + x + 1, which is increasing for all xx, so ff has an inverse gg. (a) Verify f(1)=3f(1) = 3. (b) Find f′(x)f'(x) and f′(1)f'(1). (c) Use the inverse-function rule to find g′(3)g'(3).
Show worked answer →

A 4-point inverse-derivative question.

(a) (1 point) f(1)=1+1+1=3f(1) = 1 + 1 + 1 = 3, so g(3)=1g(3) = 1.
(b) (1 point) f′(x)=3x2+1f'(x) = 3x^2 + 1, so f′(1)=3(1)+1=4f'(1) = 3(1) + 1 = 4.
(c) (2 points) g′(3)=1f′(g(3))=1f′(1)=14g'(3) = \frac{1}{f'(g(3))} = \frac{1}{f'(1)} = \frac{1}{4}.

Related dot points

Sources & how we know this