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Gradient
A gradient is a vector of partial derivatives. It describes the direction of steepest ascent for a differentiable function.
Notation
The gradient of function f is notated as ∇f. In terms of partial derivatives, the gradient of f(x1, x2, ... xn) is:
At a given point p, as long as the function f is differentiable at p, the gradient vector is:
Note the assumption; it is not negligible. For example, (xy)/(x2 + y2) is partially derivable but is itself not totally derivable at point p = [0 0]. Furthermore, it is not derivable if rotated; the basis must be orthonormal.
Also note that, technically, this is the transpose of the gradient; vectors are by default columns.
Usage
By setting a gradient to 0, critical points (local minima, local maxima, and inflections) can be calculated.
More generally, gradient descent can be used to estimate minima.