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MathidiaMathematics Academy
Differentiation
Lesson 1 of 11
Introduction

Differentiation from first principles

Objective. Derive a gradient function using the limit definition.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

The derivative is the limit of the gradient of a chord as the two points move together: f′(x) = lim(h→0) [f(x+h) − f(x)] / h.

Worked example

Differentiate f(x) = x² from first principles.

f′(x) = 2x
Illustrated card showing f′(x) = 2x
  1. 1f(x+h) − f(x) = (x+h)² − x² = 2xh + h².
  2. 2Divide by h: 2x + h.
  3. 3Let h → 0.

Answer: f′(x) = 2x

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