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
\[\mathbf{f}\]