Editing Gradient (section)

===Gradient is direction of steepest ascent===
The gradient of a function <math>f \colon \R^n \to \R</math> at point {{math|''x''}} is also the direction of its steepest ascent, i.e. it maximizes its [[directional derivative]]:

Let <math> v \in \R^n</math> be an arbitrary unit vector. With the directional derivative defined as

<math display="block">\nabla_v f (x) = \lim_{h \rightarrow 0} \frac{f(x + vh) - f(x)}{h},</math>

we get, by substituting the function <math>f(x + vh)</math> with its [[Taylor series]],

<math display="block">\nabla_v f (x) = \lim_{h \rightarrow 0} \frac{(f(x) + \nabla f \cdot vh + R) - f(x)}{h},</math>

where <math>R</math> denotes higher order terms in <math>vh</math>.

Dividing by <math>h</math>, and taking the limit yields a term which is bounded from above by the [[Cauchy-Schwarz inequality]]<ref>{{cite book |author1=T. Arens | title=Mathematik |edition=5th |publisher=Springer Spektrum Berlin |year=2022 | doi=10.1007/978-3-662-64389-1 |isbn=978-3-662-64388-4 |url = https://doi.org/10.1007/978-3-662-64389-1}}</ref>

<math display="block">|\nabla_v f (x)| = |\nabla f \cdot v| \le |\nabla f| |v| = |\nabla f|.</math>

Choosing <math>v^* = \nabla f/|\nabla f|</math> maximizes the directional derivative, and equals the upper bound

<math display="block">|\nabla_{v^*} f (x)| = |(\nabla f)^2/|\nabla f|| = |\nabla f|.</math>