Derivative (Limit Definition)
Formal definition of the derivative.
Formula
Variables
Example
For f(x) = x²: f′(x) = lim (x+h)²−x²)/h = lim (2x + h) = 2x.
How it works
This formula defines the derivative as the limiting value of the average rate of change over a shrinking interval. The quotient \((f(x+h)-f(x))/h\) measures the slope of a secant line between two nearby points on the graph, and taking \(h \to 0\) makes that secant line approach the tangent line at \(x\). It works because it captures the exact instantaneous rate of change without relying on a finite step size, so it is the foundation of all differential calculus.
When to use it
Use this definition when you need to prove that a derivative exists, derive derivative rules from first principles, or compute derivatives for functions where shortcut rules are not yet available. It is also useful in physics and engineering whenever you want an instantaneous velocity, growth rate, or sensitivity from a changing quantity.
Worked example, step by step
Find the derivative of f(x) = 1/x at x = 2 using the limit definition. Step 1: Write the difference quotient. f'(2) = lim(h→0) [(f(2+h) - f(2))/h] Step 2: Substitute the function values. f'(2) = lim(h→0) [1/(2+h) - 1/2] / h Step 3: Combine the fractions in the numerator. 1/(2+h) - 1/2 = [2 - (2+h)] / [2(2+h)] = -h / [2(2+h)] Step 4: Divide by h. [ -h / (2(2+h)) ] / h = -1 / [2(2+h)] Step 5: Take the limit as h → 0. f'(2) = lim(h→0) -1 / [2(2+h)] = -1/4 Final answer: f'(2) = -1/4
Common mistakes to avoid
- Forgetting to substitute x+h into the whole function, not just adding h to the output
- Cancelling h before factoring or simplifying, which can hide the limit or cause incorrect algebra
- Stopping before evaluating the limit, leaving the derivative as a difference quotient instead of a final number or expression
Did You Know?
This limit is the rigorous definition of a derivative — the slope of a line touching a curve at a single point.
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