Directional derivative

The directional derivative of a function at a point is its rate of change there along a specified direction, defined by a limit of difference quotients.

Gradient: The vector of a scalar function’s partial derivatives, pointing toward its steepest local increase. For differentiable functions, its dot product with a unit direction gives the directional derivative.
Euclidean vector: A quantity with magnitude and direction in Euclidean space. A direction is represented by a vector, usually normalized to unit length.
Total derivative: The linear map that gives the best first-order approximation to a function’s change near a point. Unlike one directional rate, it captures first-order change in every direction at once.
Steepest descent: An iterative optimization method that moves opposite the gradient to reduce an objective function. The gradient identifies the direction of greatest increase, so its opposite gives greatest local decrease.
Partial derivative: A derivative measuring how a multivariable function changes as one coordinate varies and the others stay fixed. It is the directional derivative along a coordinate axis.
Unit vector: A vector whose length is one. Using a unit direction makes the derivative measure change per unit distance.
Subderivative: A generalized slope used for nonsmooth functions, defined through supporting affine lower bounds. At nonsmooth points, generalized subderivatives can replace ordinary directional rates in optimization.
Directional derivative test: A criterion that uses directional rates to assess local optimality or feasible improvement. At a local minimum, every available direction must have nonnegative directional rate when that rate exists.
Fréchet derivative: A linear map that approximates a function’s change near a point, with error small relative to the displacement. It provides the general linear framework behind directional change for differentiable maps.
Limit of a function: A value that a function approaches as its input approaches a specified point. The directional derivative exists only when its difference quotient has the required limit.

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