---
title: "Differentiability and the Theorems Behind the Rules"
description: "Define the derivative as a limit, prove differentiability implies continuity, state and prove Rolle's theorem using the extreme value theorem, and derive the mean value theorem from Rolle's theorem."
canonical: https://lightmysky.com/learn/mathematics/differentiability-and-the-theorems-behind-the-rules-mt_1xLEeoA69z
source: https://lightmysky.com/learn/mathematics/differentiability-and-the-theorems-behind-the-rules-mt_1xLEeoA69z.md
retrieved: 2026-11-12
---

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# Differentiability and the Theorems Behind the Rules

Define the derivative as a limit, prove differentiability implies continuity, state and prove Rolle's theorem using the extreme value theorem, and derive the mean value theorem from Rolle's theorem.

Subject: Mathematics · Area: Calculus & Analysis · Ages 20 to 21
Page: https://lightmysky.com/learn/mathematics/differentiability-and-the-theorems-behind-the-rules-mt_1xLEeoA69z

## Ready when they can

- Prove that differentiability at a point forces continuity there
- Give a continuous function that is not differentiable and say where the definition fails
- Prove Rolle's theorem using the extreme value theorem, then prove the mean value theorem from Rolle's theorem

## Lesson: Where the tangent breaks down

The derivative at a point a is the limit of the secant slope as x approaches a. If f is differentiable at a, write f(x) minus f(a) as the difference quotient times x minus a. The quotient tends to the finite number f'(a), while x minus a tends to 0. Thus f(x) tends to f(a), so f is continuous at a. This limit is taken at the single point a, so differentiability at a says nothing about differentiability at nearby points.

*(drawing: Rewrite the output change as a product, take the limit of each factor, and the product tends to 0, so f is continuous at a.)*

**Example.** Continuity does not imply differentiability. For f(x) = |x minus 2| at 2, the left secant slopes tend to -1 and the right slopes tend to 1. The derivative limit has no single value. A cusp can make slopes diverge, and a vertical tangent does not give a finite derivative. Check the two-sided difference quotient rather than relying on a graph's smooth look.

Rolle's theorem assumes continuity on [a, b], differentiability on (a, b), and f(a) = f(b). It concludes f'(c) = 0 at some interior c. The extreme value theorem gives a maximum and a minimum on [a, b]. If both occurred only at endpoints, both would equal f(a), so f would be constant and any interior c works. Otherwise an extremum sits at an interior c. At an interior maximum, right difference quotients are at most 0 and left ones at least 0, so f'(c) = 0. A minimum works the same way.

*(drawing: For f(x) = x cubed on [-1, 2], the matching point is c = 1, inside the interval. The other root, c = -1, is an endpoint.)*

**Example.** The mean value theorem keeps Rolle's hypotheses except f(a) = f(b): continuity on [a, b] and differentiability only on (a, b). Let m = (f(b) minus f(a)) divided by (b minus a) and g(x) = f(x) minus f(a) minus m times (x minus a). Then g(a) = g(b) = 0, so Rolle gives c with g'(c) = 0, that is f'(c) = m. For f(x) = x cubed on [-1, 2], m = 3 and 3c squared = 3, so c = 1 or -1. Only c = 1 is interior.

**Recap.** Differentiability gives continuity, Rolle's theorem uses an interior extremum, and subtracting the secant line turns Rolle's theorem into the mean value theorem.

*Lesson, questions and flashcards are original text written for this dataset.*

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Continuity and Uniform Continuity](https://lightmysky.com/learn/mathematics/continuity-and-uniform-continuity-mt_plWc9raOwz)

## Optional connections

- [The Mean Value Theorem](https://lightmysky.com/learn/mathematics/the-mean-value-theorem-mt_09iApYVelx)

## Opens up

- [The Riemann Integral and When a Function Is Integrable](https://lightmysky.com/learn/mathematics/the-riemann-integral-and-when-a-function-is-integrable-mt_Ahl2MYIA4N)