---
title: "Differentiable — AP Calculus Definition & Exam Guide"
description: "A function is differentiable where its derivative exists, meaning the graph is smooth with a real tangent slope. Differentiable implies continuous, not vice versa."
canonical: "https://fiveable.me/ap-calc/key-terms/differentiable"
type: "key-term"
subject: "AP Calculus AB/BC"
unit: "Unit 5"
---

# Differentiable — AP Calculus Definition & Exam Guide

## Definition

A function is differentiable at a point if its derivative exists there, meaning the limit of the difference quotient exists and the graph has a non-vertical tangent line. On the AP exam, differentiability implies continuity, but a continuous function can fail to be differentiable (like |x| at x = 0).

## What It Is

Differentiable means the [derivative](/ap-calc/unit-10-infinite-sequences-and-series-bc-only/finding-taylor-polynomial-approximations-functions/study-guide/LszguYzKz0M6GdqTRSr6 "fv-autolink") exists. At a single point, that means the limit of the [difference quotient](/ap-calc/key-terms/difference-quotient "fv-autolink") exists there, so the graph has one well-defined, non-vertical tangent line at that point. When we say a function is differentiable on an interval, the derivative exists at every point in that interval. Visually, differentiable means smooth. No corners, no jumps, no holes, no vertical tangents.

The AP CED ([Topic 2.4](/ap-calc/unit-2/determining-when-derivatives-do-do-not-exist/study-guide/Lk6aXtIExtqciduDNGhk "fv-autolink")) cares most about the relationship between differentiability and continuity. If a function is differentiable at a point, it must be continuous there. The reverse is false. A function can be continuous at a point and still fail to be differentiable in two classic ways. First, the left-hand and right-hand limits of the difference quotient disagree, which is what happens at the corner of f(x) = |x| at x = 0. Second, the tangent line is vertical and has no slope, like f(x) = ∛x at x = 0. One more consequence worth memorizing: if a point isn't in the domain of f, it can't be in the domain of f′ either.

## Why It Matters

Differentiability lives in two places in the CED. In [Unit 2](/ap-calc/unit-2 "fv-autolink"), learning objective 2.4.A asks you to explain the relationship between differentiability and continuity, which is one of the most frequently tested logic chains in the course. In Unit 5, learning objective 5.1.A has you apply the Mean Value Theorem, and the MVT only works if its hypotheses are met. Per FUN-1.B.1, f must be [continuous](/ap-calc/key-terms/continuous "fv-autolink") on the closed interval [a, b] and differentiable on the open interval (a, b). If you skip checking differentiability, you lose justification points. Differentiability is also the silent gatekeeper for almost every big theorem later in the course, including Rolle's Theorem and the reasoning behind finding critical points.

## Connections

### Continuity (Unit 1 & Unit 2)

Differentiability is the stronger condition. Every [differentiable function](/ap-calc/key-terms/differentiable-function "fv-autolink") is continuous, but continuity alone doesn't guarantee a derivative. Think of differentiability as continuity plus smoothness. The graph not only connects, it has no corners or vertical tangents.

### Mean Value Theorem & Rolle's Theorem (Unit 5)

Both theorems require differentiability on the open interval as a hypothesis. When an FRQ says 'f is differentiable on [−6, 5],' that sentence is a green light. It's the College Board handing you permission to apply MVT, and your justification should explicitly say the hypotheses are satisfied.

### [Absolute Value Function (Unit 2)](/ap-calc/key-terms/absolute-value-function)

f(x) = |x| is the AP exam's favorite counterexample. It's continuous everywhere, but at x = 0 the [slope](/ap-calc/key-terms/slope "fv-autolink") from the left is −1 and the slope from the right is +1. Since the one-sided limits of the difference quotient disagree, there's no derivative at the corner.

### [Critical Point (Unit 5)](/ap-calc/key-terms/critical-point)

[Critical points](/ap-calc/key-terms/critical-points "fv-autolink") happen where f′(x) = 0 or where f′ does not exist. That second case is exactly a failure of differentiability, so corners and vertical tangents can be critical points too. Don't only hunt for places where the derivative equals zero.

## On the AP Exam

Multiple-choice questions test the logic directly with stems like 'Which of the following functions is both continuous and differentiable at x = 0?' or 'If a function is not differentiable at a point, the function is...'. You need to spot corners, cusps, vertical tangents, and discontinuities, and you need the one-way implication cold (differentiable → continuous, never the reverse). On FRQs, 'differentiable' usually shows up as a given condition rather than the question itself. The 2017 exam stated 'f is differentiable on the closed interval [−6, 5],' and the 2018 and 2019 exams described 'twice-differentiable' and 'differentiable' functions given by tables. That phrasing is there so you can legally apply the Mean Value Theorem or Intermediate Value Theorem. Full credit on those justifications requires you to state the hypotheses, so write something like 'since f is differentiable on (a, b) and continuous on [a, b], the MVT guarantees a value c where f′(c) equals the average rate of change.'

## Differentiable vs Continuous

These are not the same thing, and the implication only runs one direction. Differentiable at a point means continuous at that point, guaranteed. But continuous does not mean differentiable. f(x) = |x| is continuous at x = 0 yet not differentiable there because of the corner, and f(x) = ∛x is continuous at x = 0 but has a vertical tangent with no slope. A quick mental check: continuous means you can draw it without lifting your pencil, differentiable means you can draw it without lifting your pencil AND without any sharp turns.

## Key Takeaways

- A function is differentiable at a point when the limit of the difference quotient exists there, which gives the graph a single non-vertical tangent line at that point.
- If a function is differentiable at a point, it must be continuous at that point, but a continuous function can still fail to be differentiable.
- The two classic failures of differentiability are a corner, where the left-hand and right-hand limits of the difference quotient disagree (like |x| at x = 0), and a vertical tangent with no slope (like ∛x at x = 0).
- If a point is not in the domain of f, then it is not in the domain of f′ either.
- The Mean Value Theorem requires f to be continuous on the closed interval [a, b] and differentiable on the open interval (a, b), and you must state both conditions to earn the justification point.
- When an FRQ tells you a function is differentiable or twice-differentiable, that phrase is a hint that a theorem like MVT applies.

## FAQs

### What does differentiable mean in AP Calculus?

A function is differentiable at a point if its derivative exists there, meaning the limit of the difference quotient exists and the graph has a defined tangent slope. Differentiable on an interval means the derivative exists at every point in that interval.

### If a function is continuous, is it automatically differentiable?

No, and this is the most tested trap on this topic. f(x) = |x| is continuous at x = 0 but not differentiable there because the slopes from the left (−1) and right (+1) don't match. The implication only works the other way: differentiable always means continuous.

### What's the difference between continuous and differentiable?

Continuous means the graph has no breaks, jumps, or holes. Differentiable adds smoothness on top of that, so no corners, cusps, or vertical tangents either. Differentiability is the stricter condition.

### Why is f(x) = ∛x not differentiable at x = 0?

The cube root function has a vertical tangent line at x = 0, and a vertical line has no slope. The function is continuous there, but since the tangent slope doesn't exist, neither does the derivative. The CED names this as one of the two standard ways differentiability fails.

### Why do AP FRQs always say 'f is differentiable' in the problem?

It's a setup for theorems. The Mean Value Theorem requires continuity on [a, b] and differentiability on (a, b), so when the 2017 FRQ said f was differentiable on [−6, 5], that line guaranteed the MVT applied. You earn justification points by citing those hypotheses explicitly.

## Related Study Guides

- [5.1 Using the Mean Value Theorem](/ap-calc/unit-5/using-mean-value-theorem/study-guide/79sP2PXcyvRvBsjb3HRq)

## Structured Data

```json
{"@context":"https://schema.org","@graph":[{"@type":"LearningResource","@id":"https://fiveable.me/ap-calc/key-terms/differentiable#resource","name":"Differentiable — AP Calculus Definition & Exam Guide","url":"https://fiveable.me/ap-calc/key-terms/differentiable","learningResourceType":"Concept explainer","educationalLevel":"AP / High School","about":{"@id":"https://fiveable.me/ap-calc/key-terms/differentiable#term"},"audience":{"@type":"EducationalAudience","educationalRole":"student"},"dateModified":"2026-06-12T22:52:42.795Z","isPartOf":{"@type":"Collection","name":"AP Calculus AB/BC Key Terms","url":"https://fiveable.me/ap-calc/key-terms"},"publisher":{"@type":"Organization","name":"Fiveable","url":"https://fiveable.me"}},{"@type":"DefinedTerm","@id":"https://fiveable.me/ap-calc/key-terms/differentiable#term","name":"Differentiable","description":"A function is differentiable at a point if its derivative exists there, meaning the limit of the difference quotient exists and the graph has a non-vertical tangent line. On the AP exam, differentiability implies continuity, but a continuous function can fail to be differentiable (like |x| at x = 0).","url":"https://fiveable.me/ap-calc/key-terms/differentiable","inDefinedTermSet":{"@type":"DefinedTermSet","name":"AP Calculus AB/BC Key Terms","url":"https://fiveable.me/ap-calc/key-terms"}},{"@type":"FAQPage","mainEntity":[{"@type":"Question","name":"What does differentiable mean in AP Calculus?","acceptedAnswer":{"@type":"Answer","text":"A function is differentiable at a point if its derivative exists there, meaning the limit of the difference quotient exists and the graph has a defined tangent slope. Differentiable on an interval means the derivative exists at every point in that interval."}},{"@type":"Question","name":"If a function is continuous, is it automatically differentiable?","acceptedAnswer":{"@type":"Answer","text":"No, and this is the most tested trap on this topic. f(x) = |x| is continuous at x = 0 but not differentiable there because the slopes from the left (−1) and right (+1) don't match. The implication only works the other way: differentiable always means continuous."}},{"@type":"Question","name":"What's the difference between continuous and differentiable?","acceptedAnswer":{"@type":"Answer","text":"Continuous means the graph has no breaks, jumps, or holes. Differentiable adds smoothness on top of that, so no corners, cusps, or vertical tangents either. Differentiability is the stricter condition."}},{"@type":"Question","name":"Why is f(x) = ∛x not differentiable at x = 0?","acceptedAnswer":{"@type":"Answer","text":"The cube root function has a vertical tangent line at x = 0, and a vertical line has no slope. The function is continuous there, but since the tangent slope doesn't exist, neither does the derivative. The CED names this as one of the two standard ways differentiability fails."}},{"@type":"Question","name":"Why do AP FRQs always say 'f is differentiable' in the problem?","acceptedAnswer":{"@type":"Answer","text":"It's a setup for theorems. The Mean Value Theorem requires continuity on [a, b] and differentiability on (a, b), so when the 2017 FRQ said f was differentiable on [−6, 5], that line guaranteed the MVT applied. You earn justification points by citing those hypotheses explicitly."}}]},{"@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"AP Calculus AB/BC","item":"https://fiveable.me/ap-calc"},{"@type":"ListItem","position":2,"name":"Key Terms","item":"https://fiveable.me/ap-calc/key-terms"},{"@type":"ListItem","position":3,"name":"Unit 5","item":"https://fiveable.me/ap-calc/unit-5"},{"@type":"ListItem","position":4,"name":"Differentiable"}]}]}
```