Derivative

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable).

Description

Derivatives are a fundamental tool of calculus.

For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time is advanced.

The derivative of a function of a single variable at a chosen input value is the slope of the tangent line to the graph of the function at that point. This means that it describes the best linear approximation of the function near that input value.

For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable.

Generalization

Derivatives may be generalized to functions of several real variables.

In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function.

The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables.

It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.

Differentiation

The process of finding a derivative is called differentiation.

Antidifferentiation

The reverse process is called antidifferentiation. The fundamental theorem of calculus states that antidifferentiation is the same as integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.

See also

• Applications of derivatives
• Automatic differentiation
• Calculus
• Dependent variable
• Differentiability class
• Differential equation
• Differentiation rules
• Differintegral
• Finite difference method
• Fractal derivative
• Function (mathematics)
• Generalizations of the derivative
• Gradient
• Hasse derivative
• History of calculus
• Independent variable
• Infinitesimal
• Integral
• Linearization
• Mathematical analysis
• Multiplicative inverse
• Numerical differentiation
• Rate (mathematics)
• Radon–Nikodym theorem
• Schwarzian derivative
• Sensitivity to change
• Symmetric derivative

External links

• Derivative @ Wikipedia