Partial Derivatives & the Gradient · Superchallenge problem

Let $f(x,y,z) = x + 2y + 3z^2$. Among all unit vectors $\mathbf{u}$ tangent to the sphere $x^2 + y^2 + z^2 = 3$ at the point $(1,1,1)$, find the largest possible value of the directional derivative $D_{\mathbf{u}}f(1,1,1)$.