THEOREM 9—The Directional Derivative Is a Dot Product
If $f(x, y)$ is differentiable in an open region containing $P_0(x_0, y_0)$, then
$$ \left( \frac{df}{ds} \right)_{\vec{u}} = \nabla f|_{P_0} \cdot \vec{u}. $$
(4)
the dot product of the gradient $\nabla f$ at $P_0$ with the vector $\vec{u}$. In brief, $D_\vec{u} f = \nabla f \cdot \vec{u}$.
2. Find the derivative of the function $f(x, y, z) = 3e^x \cos yz$ at $P(0, 0, 0)$ in the direction of $\vec{u} = 2\vec{i} + \vec{j} - 2\vec{k}$.
