The Definition of the Derivative
The derivative is the instantaneous rate of change of a function — the limit of the average rate of change (slope of a secant line) as the interval shrinks to a point.
From secant slopes to the tangent slope
The average rate of change of f over [a, a+h] is (f(a+h) - f(a)) / h — the slope of the secant line through those two points.
The derivative at a, f'(a), is what that slope approaches as h → 0: f'(a) = lim(h→0) [f(a+h) - f(a)] / h. Geometrically, it's the slope of the tangent line to the curve at x = a.
f'(a) also equals lim(x→a) [f(x) - f(a)] / (x - a), an equivalent form using x instead of a+h.
Worked Example
Use the limit definition to find f'(x) for f(x) = x².
- f'(x) = lim(h→0) [f(x+h) - f(x)] / h = lim(h→0) [(x+h)² - x²] / h
- Expand: (x+h)² - x² = x² + 2xh + h² - x² = 2xh + h²
- Divide by h: (2xh + h²)/h = 2x + h (valid for h≠0, which is fine since h→0 means h never actually equals 0)
- Take the limit as h→0: 2x + 0 = 2x
Answer: f'(x) = 2x
Differentiability implies continuity (not the reverse)
If f is differentiable at a point, it must be continuous there — you can't have a well-defined tangent slope at a break in the graph.
The converse is false: a function can be continuous but not differentiable, most commonly at a sharp corner (like |x| at x=0) or a vertical tangent, where the left- and right-hand difference quotients disagree or blow up.
Key terms (2)
- Difference quotient
- [f(a+h)-f(a)]/h — the average rate of change over an interval of width h.
- Differentiable
- A function for which the derivative (limit of the difference quotient) exists at a point.
Practice Quiz
Using the limit definition, find f'(x) for f(x) = 3x + 5.