Function of several real variables
Partial derivatives

Partial derivatives can be defined with respect to each variable:

Partial derivatives themselves are functions, each of which represents the rate of change of f parallel to one of the x, x, …, x axes at all points in the domain (if the derivatives exist and are continuous—see also below). A first derivative is positive if the function increases along the direction of the relevant axis, negative if it decreases, and zero if there is no increase or decrease. Evaluating a partial derivative at a particular point in the domain gives the rate of change of the function at that point in the direction parallel to a particular axis, a real number.

For real-valued functions of a real variable, y = f (x), its ordinary derivative dy / dx is geometrically the gradient of the tangent line to the curve y = f (x) at all points in the domain. Partial derivatives extend this idea to tangent hyperplanes to a curve.

The second order partial derivatives can be calculated for every pair of variables:

Geometrically, they are related to the local curvature of the function's image at all points in the domain. At any point where the function is well-defined, the function could be increasing along some axes, and/or decreasing along other axes, and/or not increasing or decreasing at all along other axes.

This leads to a variety of possible stationary points: global or local maxima, global or local minima, and saddle points —the multidimensional analogue of inflection points for real functions of one real variable. The Hessian matrix is a matrix of all the second order partial derivatives, which are used to investigate the stationary points of the function, important for mathematical optimization.

In general, partial derivatives of higher order p have the form:

where p, p, …, p are each integers between 0 and p such that p + p + ⋯ + p = p, using the definitions of zeroth partial derivatives as identity operators:

The number of possible partial derivatives increases with p, although some mixed partial derivatives (those with respect to more than one variable) are superfluous, because of the symmetry of second order partial derivatives. This reduces the number of partial derivatives to calculate for some p.
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