Gradient and a normalized direction

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Problem

Let $f(x,y)=x^2y+e^y$ . Compute $\nabla f(1,0)$ and the directional derivative there in the direction from $(1,0)$ toward $(4,4)$ .

Reveal answer or reference solution

\nabla f=(2xy,x^2+e^y),\qquad \nabla f(1,0)=(0,2)

The displacement direction is $(3,4)$ , whose unit vector is $(3/5,4/5)$ . Therefore

D_uf(1,0)=\nabla f(1,0)^{\mathsf T}u=\frac85

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Gradient and a normalized direction

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