Let P be a point where the gradient does not vanish. Then, the directional derivative of $f$ in the direction $\mathbf{u}$ is given by

The product of the positive quantities $∥\nabla f∥$ and $\|\cos \left(\mathrm{\θ}\right)\|$ is maximal when $\mathrm{\θ}\=0$ so that $\cos \left(\mathrm{\θ}\right)\=1$. Hence, when u is along $\nabla f$, the rate of change in $f$ is maximal. The gradient direction is therefore the direction of maximal increase in $f$.