CalcRef

Derivatives · Basic Rules

Limit Definition of Derivative

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

The derivative of f at x is defined as the limit of the difference quotient as h approaches 0.

Worked examples

Find f'(x) for f(x) = x² using the limit definition.
  1. f(x+h) = (x+h)² = x² + 2xh + h²
  2. [f(x+h) - f(x)]/h = (2xh + h²)/h = 2x + h
  3. lim(h→0) (2x + h) = 2x

Answer: f'(x) = 2x

Find f'(x) for f(x) = 1/x using the limit definition.
  1. f(x+h) = 1/(x+h)
  2. [1/(x+h) - 1/x]/h = [x - (x+h)]/(hx(x+h)) = -h/(hx(x+h)) = -1/(x(x+h))
  3. lim(h→0) -1/(x(x+h)) = -1/x²

Answer: f'(x) = -1/x²

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