The derivative of a function f at a point x is the limit of the ratio of the increment of the function Δf to the increment of the argument Δx as Δx tends to zero. It is denoted f'(x) or df/dx. If a function describes distance s(t), its first derivative s'(t) is instantaneous speed, and the second s''(t) is acceleration. Differentiation rules (sum, product, chain rule) allow finding derivatives analytically without directly computing limits.
History
In the 1660s, Newton developed his method of fluxions, and Leibniz in the 1670s created an independent formalism of differentials dx. A long dispute over priority stimulated the development of the British and continental schools. Augustin-Louis Cauchy in the 1820s gave a rigorous ε-δ definition of the limit, on which the modern definition of the derivative is based. Riemann introduced the concept of the integral, completing the theory.
How it works
Differentiation is a linear operation: (f+g)' = f'+g', (cf)' = cf'. Formulas for derivatives of elementary functions (power, trigonometric, exponential) are derived from the limit definition. For composite functions, the chain rule is applied: (f(g(x)))' = f'(g(x))·g'(x). The apparatus of derivatives allows solving optimization problems (setting the derivative equal to zero), constructing approximations, and studying the dynamics of physical systems through Newton's laws, which reduce to differential equations.
In practice
Analyzing population growth, finding the optimal price to maximize profit, and designing smooth highway curves—everywhere the derivative is needed.
Fun fact
The rate of growth of the planet's population at a given moment is the derivative of the function 'total number of inhabitants' with respect to time. Economists use the derivative for the concept of marginal cost—how much costs will increase when producing one more unit of output.
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