The Derivative — Instantaneous Rate of Change
The derivative of f at x = a is:
f'(a) = lim(h→0) [f(a+h) - f(a)] / h
What it means: The slope of the tangent line to f(x) at the point x = a. The instantaneous rate of change.
Equivalent form:
f'(a) = lim(x→a) [f(x) - f(a)] / (x - a)
When the derivative doesn't exist:
Corner or cusp (sharp point)
Vertical tangent line
Discontinuity
Notation: f'(x), dy/dx, d/dx[f(x)], y' — all mean the same thing.
The AP exam will ask you to use the limit definition. Know it cold. Even though shortcut rules exist, the definition question appears every year.
Reference:
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