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MathidiaMathematics Academy
Differentiation
Lesson 31 of 75
Introduction

Differentiation from first principles

Objective. Derive a gradient function from the limiting chord gradient.

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  2. 2. Worked example
  3. 3. Practice
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Learn: the key idea

Differentiation from first principles finds a gradient function by taking the limit of the gradient of a chord as the two points on the curve move infinitely close together. The formula is f'(x) = lim(h→0) [f(x+h) − f(x)]/h, representing the gradient between (x, f(x)) and (x+h, f(x+h)) as h shrinks to zero. For f(x) = x², we get [(x+h)² − x²]/h = [2xh + h²]/h = 2x + h, which tends to 2x as h → 0, confirming the familiar derivative. This method justifies the power rule and all other differentiation rules rather than assuming them.

Worked example

Differentiate f(x) = x² + 3x from first principles.

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Illustrated card showing 3
  1. 1Form the difference quotient: [f(x+h) − f(x)]/h = [(x+h)² + 3(x+h) − x² − 3x]/h.
  2. 2Expand the numerator: 2xh + h² + 3h.
  3. 3Divide by h: 2x + h + 3.
  4. 4Let h → 0 to get f'(x) = 2x + 3.

Answer: f'(x) = 2x + 3

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Common mistake. Students often substitute h = 0 too early, before simplifying the fraction, which gives 0/0 and no information. The algebraic expansion and cancellation of h must happen first, and only then can h be set to zero.

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