Answer
$${D_{\bf{u}}}f\left( {2, - 2,1} \right) = - \frac{1}{2}$$
Work Step by Step
$$\eqalign{
& f\left( {x,y,z} \right) = 1 + \sin \left( {x + 2y - z} \right);\,\,\,\,P\left( {\frac{\pi }{6},\frac{\pi }{6}, - \frac{\pi }{6}} \right);\,\,\,\,{\bf{u}} = \left\langle {\frac{1}{3},\frac{2}{3},\frac{2}{3}} \right\rangle \cr
& {\bf{u}} = \sqrt {{{\left( {\frac{1}{3}} \right)}^2} + {{\left( {\frac{2}{3}} \right)}^2} + {{\left( {\frac{2}{3}} \right)}^2}} = 1 \cr
& {\bf{u}}{\text{ is a unit vector}} \cr
& \cr
& {\text{The gradient of }}f\left( {x,y} \right){\text{ is}} \cr
& {f_x}\left( {x,y,z} \right) = \cos \left( {x + 2y - z} \right) \cr
& {f_y}\left( {x,y,z} \right) = 2\cos \left( {x + 2y - z} \right) \cr
& {f_z}\left( {x,y,z} \right) = - \cos \left( {x + 2y - z} \right) \cr
& \nabla f\left( {x,y,z} \right) = \cos \left( {x + 2y - z} \right)\left( {{\bf{i}} + 2{\bf{j}} - {\bf{k}}} \right) \cr
& \cr
& {\text{Calculate the gradient at the point }}P\left( {\frac{\pi }{6},\frac{\pi }{6}, - \frac{\pi }{6}} \right) \cr
& \nabla f\left( {\frac{\pi }{6},\frac{\pi }{6}, - \frac{\pi }{6}} \right) = \cos \left( {\frac{\pi }{6} + \frac{\pi }{3} + \frac{\pi }{6}} \right)\left( {{\bf{i}} + 2{\bf{j}} - {\bf{k}}} \right) \cr
& \nabla f\left( {\frac{\pi }{6},\frac{\pi }{6}, - \frac{\pi }{6}} \right) = - \frac{1}{2}\left( {{\bf{i}} + 2{\bf{j}} - {\bf{k}}} \right) \cr
& \nabla f\left( {\frac{\pi }{6},\frac{\pi }{6}, - \frac{\pi }{6}} \right) = \left\langle { - \frac{1}{2}, - 1,\frac{1}{2}} \right\rangle \cr
& \cr
& {\text{Computing the directional derivatives of }}f{\text{ at }}\left( {\frac{\pi }{6},\frac{\pi }{6}, - \frac{\pi }{6}} \right) \cr
& \operatorname{in} {\text{ the direction of the vector }}{\bf{u}} = \left\langle {\frac{1}{3},\frac{2}{3},\frac{2}{3}} \right\rangle \cr
& {D_{\bf{u}}}f\left( {a,b,c} \right) = \nabla f\left( {a,b,c} \right) \cdot {\bf{u}} \cr
& {D_{\bf{u}}}f\left( {2, - 2,1} \right) = \left\langle { - \frac{1}{2}, - 1,\frac{1}{2}} \right\rangle \cdot \left\langle {\frac{1}{3},\frac{2}{3},\frac{2}{3}} \right\rangle \cr
& {D_{\bf{u}}}f\left( {2, - 2,1} \right) = - \frac{1}{6} - \frac{2}{3} + \frac{1}{3} \cr
& {D_{\bf{u}}}f\left( {2, - 2,1} \right) = - \frac{1}{2} \cr} $$
