Directional Derivatives — Question 2

Question 2 – Solution

Strategy. Subtract endpoint coordinates in the correct order before normalizing.

Step 1: Direction $$\overrightarrow{PQ}=\left\langle 3,4\right\rangle,\qquad u=\left\langle 3/5,4/5\right\rangle.$$

Step 2: Gradient $$\nabla g=\left\langle\frac{2x}{x^2+y^2},\frac{2y}{x^2+y^2}\right\rangle,$$ so $$\nabla g(1,-2)=\left\langle 2/5,-4/5\right\rangle.$$ Therefore $$D_{u}g(P)=\frac 25\frac 35-\frac 45\frac 45 =\boxed{-\frac 25}.$$

Step 3: Interpret The negative sign means $g$ decreases initially toward $Q$. This is consistent with the initial motion having a component toward smaller distance from the origin.