Editing Gradient (section)

====Linear approximation to a function====
The best [[linear approximation]] to a function can be expressed in terms of the gradient, rather than the derivative. The gradient of a [[function (mathematics)|function]] <math>f</math> from the Euclidean space <math>\R^n</math> to <math>\R</math> at any particular point <math>x_0</math> in <math>\R^n</math> characterizes the best [[linear approximation]] to <math>f</math> at <math>x_0</math>. The approximation is as follows:

<math display="block">f(x) \approx f(x_0) + (\nabla f)_{x_0}\cdot(x-x_0)</math>

for <math>x</math> close to <math>x_0</math>, where <math>(\nabla f)_{x_0}</math> is the gradient of <math>f</math> computed at <math>x_0</math>, and the dot denotes the dot product on <math>\R^n</math>. This equation is equivalent to the first two terms in the [[Taylor series#Taylor series in several variables|multivariable Taylor series]] expansion of <math>f</math> at <math>x_0</math>.