Derivatives

A derivative describes how a quantity changes with respect to another quantity.

What is a derivative?

Formally, the derivative of a function f at a point x is the limit of the average rate of change over an interval as that interval shrinks to zero — geometrically, it’s the slope of the line tangent to the function’s graph at that point.

Basic formula

f’(x) = lim(h→0) [f(x+h) − f(x)] / h. In practice, derivatives are usually computed with rules rather than the limit directly — the power rule, product rule, quotient rule, and chain rule cover most functions encountered in coursework.

Example

For f(x) = x³, the power rule gives f’(x) = 3x². At x = 2, the instantaneous rate of change is f’(2) = 12.

Common mistakes

Forgetting the chain rule when differentiating a composite function; confusing the derivative (a rate of change) with the function’s value; dropping constants incorrectly when applying the power rule (the derivative of a constant is 0, not the constant itself).

Integrals

Questions about derivatives

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