What is a derivative?
The instantaneous rate of change of a function — the slope of the curve at a single point, with no averaging.
The idea
If a function tells you where something is, the derivative tells you how fast it's moving. For a position function s(t), the derivative s'(t) is velocity. For a population P(t), P'(t) is growth rate.
The clever part: at any single instant, "rate of change" is meaningless in everyday language — nothing has elapsed. Calculus solves this by averaging over a tiny interval, then asking what happens as the interval shrinks to zero.
The definition
The derivative of f at the point x is the limit of the average rate of change over intervals that shrink to zero:
f'(x) = limh → 0 [ f(x + h) − f(x) ] / hIf this limit exists, f is said to be differentiable at x. Geometrically, f'(x) is the slope of the tangent line to the graph of f at the point (x, f(x)).
Worked example
Find the derivative of f(x) = x².
f(x + h) = (x + h)² = x² + 2xh + h² f(x + h) − f(x) = 2xh + h² [f(x + h) − f(x)] / h = 2x + h limh → 0 (2x + h) = 2xSo f'(x) = 2x. At x = 3 the tangent slope is 6. At x = 0 the slope is 0 — the curve momentarily flattens at the origin, just as the parabola does.
Flashcard · Unit 2
Bring the exponent down, subtract one.
Check your understanding
If f(x) = 5x² + 3x, what is f'(x)?