Correct Expression of Material Derivative in Continuum Physics

The material derivative is important in continuum physics. It shows that $\frac{d }{dt}=\frac{\partial }{\partial t}+(\bm v\cdot \bm \nabla)$ is a wrong expression. The correct expression $ \frac{d (:)}{dt}=\frac{\partial }{\partial t}(:)+\bm v\cdot [\bm \nabla (:)]$ is to be formulated.