Directional Derivatives — Question 6

Question 6 – Solution

Strategy. The same gradient-dot-unit-vector formula applies in three dimensions.

Step 1: Direction $$\overrightarrow{PQ}=\left\langle-3,2,2\right\rangle,\qquad \|\overrightarrow{PQ}\|=\sqrt{17},\qquad u=\frac 1{\sqrt{17}}\left\langle-3,2,2\right\rangle.$$

Step 2: Gradient $$F_x=e^{yz},\qquad F_y=xze^{yz},\qquad F_z=xye^{yz}+2z.$$ At $(2,0,1)$, $$\nabla F(P)=\left\langle 1,2,2\right\rangle.$$

Step 3: Rate $$D_{u}F(P)=\frac{-3+4+4}{\sqrt{17}} =\boxed{\frac 5{\sqrt{17}}}.$$ Because $u$ is dimensionless and unit length, the result measures change in $F$ per unit spatial distance.