Derivative

A derivative of a function $f(x)$ at a point $x_0$ is defined as the limit

$f'(x_0) = \lim_{h \to 0} \frac{f(x_0+h) - f(x_0)}{h}$

provided the limit exists. Geometrically it is the slope of the tangent line at $(x_0, f(x_0))$; physically it is the instantaneous rate of change of the quantity represented by $f$.

Derivatives are linear (the derivative of a sum is the sum of derivatives), and a small set of rules — power, product, quotient, chain — let you differentiate most elementary functions mechanically without going back to the limit definition every time.

Derivatives are foundational to optimisation (finding maxima and minima), to physics (velocity is the derivative of position, acceleration of velocity), to machine learning (gradient descent), and to economics (marginal cost / revenue).