## 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
$\mathbf{f}$