Question

Find the directional derivative of f(x, y, z) = xy + yz + zx at p(1, −1, 7) in the direction from p to q(2, 4, 5). duf(1, −1, 7)

Answer 1

$f(x,y,z)=xy+yz+zx$

$\implies\nabla f(x,y,z)=(y+z,x+z,y+x)$

$\vec{pq}=\mathbf q-\mathbf p=(2-1,4-(-1),5-7)=(1,5,-2)$

$\|\vec{pq}\|=\sqrt{30}$

$D_{\vec{pq}}f(1,-1,7)=\nabla f(1,-1,7)\cdot\dfrac{(1,5,-2)}{\sqrt{30}}=\dfrac{(6,8,0)\cdot(1,5,-2)}{\sqrt{30}}=\dfrac{46}{\sqrt{30}}$