[tex]g(x,y,z)=(x+2y+3z)^{3/2}[/tex]
[tex]\implies\nabla g(x,y,z)=\dfrac32\left((x+2y+3z)^{1/2},2(x+2y+3z)^{1/2},3(x+2y+3z)^{1/2}\right)[/tex]
In the direction of [tex]\mathbf v=4\,\mathbf j-\mathbf k=(0,4,-1)[/tex], with norm [tex]\sqrt{17}[/tex], we have the directional derivative
[tex]D_{\mathbf v}g(5,4,4)=\nabla g(5,4,4)\cdot\dfrac{(0,4,-1)}{\sqrt{17}}=\dfrac1{\sqrt{17}}\left(\dfrac{15}2,15,\dfrac{45}2\right)\cdot(0,4,-1)=\dfrac{75}{2\sqrt{17}}[/tex]
