[tex]f(x,y,z)=xy+z^3[/tex]
has gradient
[tex]\nabla f(x,y,z)=\langle y,x,3z^2\rangle[/tex]
The derivative of [tex]f(x,y,z)[/tex] at [tex]P[/tex] in the direction of a vector [tex]\vec v[/tex] is
[tex]D_{\vec u}f(P)=\nabla f(P)\cdot\dfrac{\vec u}{\|\vec u\|}[/tex]
In this case, [tex]\vec u[/tex] is the vector pointing from [tex]P[/tex] to the origin, which is
[tex]\vec u=\langle0,0,0\rangle-\langle4,-7,-2\rangle=\langle-4,7,2\rangle[/tex]
and has norm
[tex]\|\vec u\|=\sqrt{(-4)^2+7^2+2^2}=\sqrt{69}[/tex]
Then the derivative of [tex]f[/tex] at [tex]P[/tex] in the direction of [tex]\vec u[/tex] is
[tex]D_{\vec u}f(P)=\langle-7,4,12\rangle\cdot\dfrac{\langle-4,7,2\rangle}{\sqrt{69}}=\dfrac{80}{\sqrt{69}}[/tex]
