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| A '''gradient''' is a vector of partial derivatives. It describes the direction of steepest ascent for a differentiable function. | A '''gradient''' is a vector of [[Calculus/PartialDerivative|partial derivatives]]. It describes the direction of steepest ascent for a [[Calculus/Derivative|differentiable]] function. |
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| The gradient of function ''f'' is notated as ''∇f''. In terms of [[Calculus/PartialDerivatives|partial derivatives]], the gradient of ''f(x,,1,,, x,,2,,, ... x,,n,,)'' is: | The gradient of function ''f'' is notated as ''∇f''. In terms of [[Calculus/PartialDerivative|partial derivatives]], the gradient of ''f(x,,1,,, x,,2,,, ... x,,n,,)'' is: |
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| ---- == Usage == By setting a gradient to 0, critical points (local minima, local maxima, and inflections) can be calculated. More generally, [[Calculus/GradientDescent|gradient descent]] can be used to estimate minima. |
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.
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.