## Proof That the Component of Grad f in the Direction of a Vector Equals the Directional Derivative of f in that Direction

The component of\[\nabla f\]

in the direction of a vector \[\mathbf{v}\]

is \[\nabla f \cdot \frac{\mathbf{v}}{|\mathbf{v}|} =|{\nabla f}| \cos \theta\]

where \[\theta\]

is the angle between \[\nabla f\]

and \[\mathbf{v}\]

. Note \[\frac{\mathbf{v}}{|\mathbf{v}|}\]

is a unit vector.This is exactly the directional derivative of

\[f\]

in the direction of \[\mathbf{f}\]