Straight vs Partial derivative Does it make sense to write $\frac{d}{dx}u(x,t)$ or can one only write $\frac{\partial}{\partial x}u(x,t)$?
For convention's sake, I'll pretend you asked about $\frac{d}{dt}$ and $\frac{\partial}{\partial t}$. It's the same exact question this way, and you'll see the reason for the change in notation afterwards.

Getting to the point, say you have a function of two variables $$u:\Bbb R^2\to\Bbb R^k\atop (x,t)\mapsto u(x,t)$$

For example, let $k=1$ and we have a scalar field (which we can think of as dependent on "position" $x$ and time $t$, if we wish). Now consider a real function $x(t)$ (forgive me for using the same letter $x$ for a completely new term). This determines a path through $\Bbb R^2: \gamma (t) =(x(t),t)$. We can compose the two functions to get a real function of $t$: $u\circ\gamma$. Now, this is a sort of liberal use of notation, but we can "identify" $u$ with $u\circ\gamma$. Now the derivative of $u$ is $$\frac{d}{dt}u = \frac{d}{dt}u(x(t),t) = \frac{\partial u }{\partial x}x'(t) + \frac{\partial u}{\partial t}$$

You can see for yourself that both $\frac{\partial}{\partial t}$ and $\frac{d}{dt}$ are applied to $u$, to mean different things. Again, this is artificious notation (at least, I haven't seen it formalized), but it's usually just comfortable to write this way, so many people do.
Actually, the $\frac{dy}{dx}$ maybe considered as a quotient of two differentials, since $dy= y' \cdot dx$, whereas $\frac{\partial y}{\partial x}$ is an indivisible symbol.

According to this, it's convinient to write such type of equations $\frac{d(f(g(x))}{dx}= \frac{\partial{f}}{\partial{g}} \cdot \frac{\partial{g}}{\partial{x}}$