I'm studying fluid mechanics, and I got the impression that the material derivative is nothing more than "differentiating along a path" and so I got confused on why do we need it. Basically, let $D\subset \mathbb{R}^3$ be the region containing the fluid and let $f : D\times \mathbb{R}\to \mathbb{R}$ be a time dependent function on $D$.

Suppose then $\gamma : I\subset \mathbb{R}\to D$ is a trajectory on the fluid. The material derivative of $f$ along $\gamma$ is defined by

$$\dfrac{D}{Dt}f(\gamma(t),t) = \dfrac{\partial f}{\partial t}(\gamma(t),t) + (\mathbf{u}\cdot \nabla)f(\gamma(t),t)$$

Where $\mathbf{u}$ is the spatial velocity field of the fluid. But that expression is nothing more nothing less than simply differentiating $f(\gamma(t),t)$ with respect to $t$, or better, differentiating the function $f\circ (\gamma, I)$ where $I$ is the identity in $\mathbb{R}$.

Indeed we have

$$\dfrac{d}{dt}f(\gamma(t),t) = \nabla f(\gamma(t),t)\cdot \gamma'(t) + \dfrac{\partial f}{\partial t}(\gamma(t),t) = \dfrac{D}{Dt} f(\gamma(t),t)$$

So since $\dfrac{d}{dt}= \dfrac{D}{Dt}$ why do we need the material derivative? Why do we define it, since in truth it is just the well know derivative of a composition just? What are the advantages of defining it?