deriválni

Deriválni is the Hungarian term for the mathematical operation of differentiation, i.e., computing the derivative of a function with respect to a variable. The derivative measures the instantaneous rate of change of the function and can be interpreted as the slope of the tangent line to the function’s graph at a given point.

Notation and basic definition

For a function f of a real variable x, the derivative is denoted f'(x) or df/dx and is defined by the limit

f'(x) = lim (h → 0) [f(x + h) − f(x)] / h,

when this limit exists. In higher dimensions, partial derivatives ∂f/∂x measure the rate of change with respect to one variable while keeping others fixed.

Rules and common forms

Derivatives obey several standard rules:

- Sum rule: d/dx [u(x) + v(x)] = u'(x) + v'(x)

- Product rule: d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)

- Chain rule: d/dx [g(f(x))] = g'(f(x)) · f'(x)

For common functions: d/dx x^n = n x^{n−1} (n constant); d/dx sin x = cos x; d/dx e^x = e^x; d/dx ln x = 1/x.

Higher-order and partial derivatives

The second derivative is f''(x) = d/dx [f'(x)], giving the rate of the rate of change. Higher-order derivatives and mixed partial derivatives ∂^2f/∂x∂y are used in physics, engineering and optimization. In multivariable calculus, ∂f/∂x, ∂f/∂y describe how f changes with each variable.

Historical context and applications

The concept emerged in the works of Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century, providing a formal framework for instantaneous change, slopes, and rates. Today, differentiation is fundamental in physics (velocity, acceleration), economics (marginal analysis), biology, and computer science, and is complemented by numerical methods for functions not easily differentiable analytically.