Differentiation
Lesson 1 of 11
Differentiation from first principles
Objective. Derive a gradient function using the limit definition.
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.
- 1f(x+h) − f(x) = (x+h)² − x² = 2xh + h².
- 2Divide by h: 2x + h.
- 3Let h → 0.
Answer: f′(x) = 2x
Practice
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