Chapter 12, Problem 65RE

To determine

To compute: The gradient of $f\left(x,y,z\right)=1+\sin \left(x+2y-z\right)$, then use it to evaluate at the point $P\left(\frac{\pi }{6},\frac{\pi }{6},-\frac{\pi }{6}\right)$ and find the directional derivative of the function at the point $P\left(\frac{\pi }{6},\frac{\pi }{6},-\frac{\pi }{6}\right)$ in the direction of the vector $u=\langle \frac{1}{3},\frac{2}{3},\frac{2}{3}\rangle$.