derivative

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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

The idea emerged in the 17th century simultaneously with Isaac Newton (as 'fluxion'—rate of change) and Gottfried Leibniz (geometric approach through tangents). Later, Augustin-Louis Cauchy gave a rigorous definition through limits.

How it works

It's like high-speed photography: instead of a blurred average motion over a second, we see an instantaneous position. Mathematically, it's the limit of the ratio of a very small increment of the function to an equally small increment of the argument.

💡 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.

Let a function f: U → ℝ be given, where U is an open set on the real line. The derivative of f at a point x₀ is defined as f'(x₀) = lim_{h→0} [f(x₀+h)−f(x₀)]/h, if the limit exists. Geometrically, it is the tangent of the angle of inclination of the tangent line to the graph. The class C¹ means that the derivative is continuous. Generalizations: directional derivative and Fréchet derivative for functions of several variables, partial derivatives. The relationship with the integral is given by the Fundamental Theorem of Calculus. The derivative is the basis of linear approximations (differential) and numerical methods (Newton's method).

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.

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

f'(x) is the derivative of the function f at the point x; h is an infinitesimal increment of the argument, tending to zero; the fraction is the difference quotient, the tangent of the slope of the secant.

\frac{d}{dx} x^n = n x^{n-1}

n is a constant exponent, can be any real number; x is the variable; rule: the exponent is brought out as a factor, the power decreases by one.

Open problems & fun facts

Existence of nowhere differentiable continuous functions (Weierstrass function)—what does 'smoothness' mean for natural processes?; Development of efficient algorithms for automatic differentiation for machine learning

💡 The derivative of the exponential is unique: (eˣ)' = eˣ. Moreover, any function proportional to eˣ has the same shape and growth rate—the only family of eigenfunctions of the differentiation operator with a real eigenvalue.

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derivative

History

How it works

History

How it works

History

How it works

Open problems & fun facts

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