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.

  • 2 worked examples
  • 24 in Derivatives

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

1 more worked examplePremium

Find f'(x) for f(x) = 1/x using the limit definition.

Step-by-step solution locked.

Unlock the full working. CalcRef Premium opens the step-by-step solution above, and 141 worked solutions across the whole library - the intermediate lines, not just the answer. The formula statement, its conditions, its variables and the first worked example stay free on every page.
See PremiumBrowse the free reference

See all 24 Derivatives formulas on one printable sheet - the same statements and conditions as these pages, typeset and laid out for paper.

Common questions

What is the Limit Definition of Derivative?

The derivative of f at x is defined as the limit of the difference quotient as h approaches 0. It is one of the derivatives formulas in the CalcRef reference.

How do you use the Limit Definition of Derivative?

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

Practice

Quiz yourself on Derivatives

Limit Definition of Derivative is one of 24 Derivatives formulas in CalcRef, out of 136 in the library. Today's free round is 10 mixed multiple-choice questions drawn from the same dataset as this page, refreshed every day and replayable as often as you like - no account needed.

Take today's free quizAll 24 Derivatives formulas