---
title: "Differentiability and the Theorems Behind the Rules"
description: "Define the derivative as a limit, prove differentiability implies continuity, and prove Rolle and the mean value theorem. The rules used all year get their justification here."
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: 2026-09-12
---

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# Differentiability and the Theorems Behind the Rules

Define the derivative as a limit, prove differentiability implies continuity, and prove Rolle and the mean value theorem. The rules used all year get their justification here.

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 the mean value theorem from Rolle's theorem

## Lesson: Where the tangent breaks down

The derivative at a point is the limit of secant slopes as the interval collapses. Differentiability there forces continuity, and the proof is one rearrangement: write the change in output as the difference quotient times the change in input, then let the input change shrink. The quotient approaches the derivative, so the product shrinks to zero. Quote this whenever a proof needs continuity for free.

**Example.** Continuity does not imply differentiability. The absolute value function is unbroken at zero but has a corner there, so the left and right secant limits disagree and the two sided limit does not exist. Corners, cusps, and vertical tangents are the three shapes where the definition fails.

The mean value theorem needs continuity on the whole closed interval plus differentiability at every interior point. To prove it from Rolle's theorem, subtract the secant line from f: the tilted function starts and ends level, so Rolle supplies a point with zero slope, which untilts to the secant slope. For f(x) equal to x squared on [0, 4], the secant slope is 4, and f prime equal to 2x hits 4 at c equal to 2.

**Recap.** Differentiability hands you continuity, corners break the derivative limit, and tilting by the secant line turns Rolle into the mean value theorem.

## Practice

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

## Needs first

- [The Mean Value Theorem](https://lightmysky.com/learn/mathematics/the-mean-value-theorem-mt_09iApYVelx)
- [Continuity and Uniform Continuity](https://lightmysky.com/learn/mathematics/continuity-and-uniform-continuity-mt_plWc9raOwz)

## 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)