User:Mrconter1/sandbox

Example

Thus, if the function $f$ and vector $\mathbf {v}$ is defined as:

$f(\mathbf {x} ,\mathbf {y} )=x^{2}+y^{2}$

and:

$\mathbf {v} =(1,2),|\mathbf {v} |={\sqrt {1^{2}+2^{2}}}={\sqrt {5}}$

The directional derivative will be:

$\nabla _{\mathbf {v} }{f}={\frac {\partial {f}}{\partial {\mathbf {x} }}}\cdot {\frac {\mathbf {v_{x}} }{|\mathbf {v} |}}+{\frac {\partial {f}}{\partial {\mathbf {y} }}}\cdot {\frac {\mathbf {v_{y}} }{|\mathbf {v} |}}=2x\cdot {\frac {1}{\sqrt {5}}}+2y\cdot {\frac {2}{\sqrt {5}}}$