directional derivative

The directional derivative of $f$ in the direction $\vec{u}$, assuming that $\vec{u}$ is a direction vector.

$$\begin{array}{rl}{D}_{\vec{u}}f(x,y,z)& =f_x\cdot u_x+f_y\cdot u_y+f_z\cdot u_z\\ & =\nabla f(x,y,z)\cdot \frac{\vec{u}}{|\vec{u}|}\\ & =⟨f_x,f_y,f_z⟩\cdot \frac{\vec{u}}{|\vec{u}|}\end{array}$$

where $f_x$ and so on are the partial derivatives with respect to the subscripted variable.

If $\vec{u}$ is the direction vector ($|\vec{u}|=1$) of $\nabla f$ at point $P$, then:

$$D_{\vec{u}}f(x,y,z)=\nabla f\cdot \vec{u}$$