Given the following equation:
$V(x,y,z)=5x^2-3xy+xyz$
Part 1: At point $P(3,4,5)$, find the rate of change in the direction of the vector $\langle1,1,-1\rangle$
Part 2: Find the direction in which $V$ changes most rapidly at $P(3,4,5)$
Part 3: Find the maximum rate of change at $P(3,4,5)$
I think I've managed to do part 1. Here's what I've done so far - but I am clueless as to how to find the max rate of change at $P$ and the direction in which the change occurs most rapidly.
$V_x(x,y,z)=10x+y(z-3)$
$V_y(x,y,z)=x(z-3)$
$V_z(x,y,z)=xy$
Based on the above partial derivatives,
$\begin{align} \nabla V(3,4,5) &=\left.\langle10x+y(z-3),\ x(z-3),\ xy\rangle\right|_{(3,4,5)} \\ &=\langle38,6,12\rangle \end{align}$
Unit vector of $\langle1,1,-1\rangle$ is $\vec u=\langle\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}\rangle$.
Directional derivative at $(3,4,5)$ in the direction of $\langle1,1,-1\rangle$ is: $\begin{align} D_\vec uV(3,4,5) &= \nabla V(3,4,5) \cdot \vec u \\ &=\frac{32}{\sqrt{3}} \end{align}$
How should I proceed?