Vector Calculus

Maximum rate of increase is $|\nabla \phi|$. $\nabla \phi = 4x\hat{i} + 6y\hat{j} = 4\hat{i} + 6\hat{j}$. $|\nabla \phi| = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} \approx 7.211$.