The directional derivative of [tex]f[/tex] at the point [tex]\vec p[/tex] in the direction of [tex]\vec v[/tex] is
[tex]D_{\vec v}f(\vec p)=\nabla f(\vec p)\cdot\dfrac{\vec v}{\|\vec v\|}[/tex]
We have
[tex]f(x,y,z)=xe^y+ye^z+ze^x\implies\nabla f(x,y,z)=\langle e^y+ze^x,xe^y+e^z,ye^z+e^x\rangle[/tex]
With [tex]\vec p=\langle0,0,0\rangle[/tex],
[tex]\nabla f(0,0,0)=\langle1,1,1\rangle[/tex]
[tex]\vec v[/tex] has magnitude
[tex]\|\vec v\|=\sqrt{5^2+2^2+(-2)^2}=\sqrt{33}[/tex]
and so the directional derivative is
[tex]\langle1,1,1\rangle\cdot\dfrac{\langle5,2,-2\rangle}{\sqrt{33}}=\boxed{\dfrac5{\sqrt{33}}}[/tex]
