The Derivative as the Limit of the Difference Quotient
Simple Explanation
Formally, the derivative of f at x is f'(x) = lim(h→0) [f(x+h) − f(x)]/h — the limit of the average rate of change over an interval of width h, as h shrinks to zero. This "difference quotient" is the same average-rate-of-change idea, just written with the second point as x+h instead of a fixed b.
Why Do We Need It?
This is the actual formal definition of the derivative — every differentiation rule in this chapter can, in principle, be derived directly from this one definition.
See It
A parabola y=x squared, with a straight tangent line touching the curve at the single point (2,4)
Formula
The Definition of the Derivative
f'(x) = lim(h→0) [f(x+h) − f(x)] / h
The derivative of f at x is the limit of the average rate of change over a shrinking interval [x, x+h], as that interval's width h shrinks to zero — giving the exact instantaneous rate of change.
- h
- — a small change in x, which shrinks toward (but never reaches) 0
- f'(x)
- — the derivative of f at x — the instantaneous rate of change, and the slope of the tangent line there
When to use it: Whenever a derivative needs to be found directly from its formal limit definition (rather than a shortcut rule).
Worked Example
Find a derivative from the limit definition
Use the limit definition of the derivative to find f'(x) for f(x) = x².
Why Does This Work?
This is exactly the average rate of change formula, [f(b)−f(a)]/(b−a), with a=x and b=x+h (so b−a=h). As h shrinks toward 0, the second point (x+h, f(x+h)) slides along the curve toward (x, f(x)), and the secant line through them rotates to become the tangent line at x — its slope in the limit is exactly the derivative.
Real-Life Example
Defining instantaneous velocity precisely
A physicist needs a mathematically rigorous definition of "velocity at an exact instant," not just average velocity over some measured time interval.
The limit of the average velocity, [s(t+h)−s(t)]/h, as h shrinks to zero, IS the rigorous definition of instantaneous velocity — precisely this difference quotient.
Practice
Using f'(x) = 2x (found above for f(x)=x²), find f'(5).
HardCommon mistake
Substituting h=0 too early, before simplifying — this gives 0/0 (indeterminate) immediately; the expression must be factored and simplified FIRST, and only then can h=0 safely be substituted.
Quick Review
- f'(x) = lim(h→0) [f(x+h) − f(x)] / h — the formal definition of the derivative.
- This is the average rate of change formula with b=x+h, as h shrinks to 0.
- Simplify (factor and cancel h) before substituting h=0.