Answer
$-2\sqrt2$
Work Step by Step
Formula to calculate the directional derivative: $D_uf=\nabla f(x,y) \cdot
u$
or, from the Dot Product Theorem, $D_uf=║\nabla f(x,y)║║u║\cos θ$
where θ is the angle between ∇f(x,y) and the vector u.
From the given figure, we have $\theta=180^\circ+45^\circ=225^\circ$ or $\dfrac{5\pi}{4}$.
At $(2,2)$,
$D_uf (2,2)=(4)(1) \cdot \cos\dfrac{5\pi}{4}$
Thus, $D_uf (2,2)=-2\sqrt2$
