Directional Derivatives — Question 1

Question 1 – Solution

Strategy. A directional derivative uses a unit vector: $D_{u}f=\nabla f\cdot u$.

Step 1: Normalize $$\|v\|=5,\qquad u=\boxed{\left\langle 3/5,4/5\right\rangle}.$$

Step 2: Gradient $$f_x=2xy+e^{x-y},\qquad f_y=x^2-e^{x-y}.$$ At $(1,0)$, $$\nabla f(1,0)=\left\langle e,1-e\right\rangle.$$

Step 3: Dot product $$D_{u}f(1,0)=e\frac 35+(1-e)\frac 45 =\boxed{\frac{4-e}{5}}.$$ The positive value means $f$ initially increases by approximately $(4-e)/5$ output units per unit distance traveled in the specified direction.