Is it correct to use partial derivative notation even when we know that function is dependent on only one variable? This is a question about notation. I am trying to understand when it is appropriate to rewrite $ \frac{\partial f}{\partial x} $ as $ \frac{df}{dx} $ while performing a derivation.
Say we are given,
$$ f(x, y) = 4x^2 + 27y^3. $$
Let us define,
$$ u(x) = 2x. $$
Then,
$$
f(x, y) = f(u(x), y) = (u(x))^2 + 27y^3.
$$
If I now want to find the partial derivative $ \frac{\partial f(x, y)}{\partial x} $, I could do the following.
$$
\frac{\partial f(x, y)}{\partial x} = \frac{\partial f(u, y)}{\partial u} \cdot \frac{\partial u(x)}{\partial x}
$$
Is the above expression correct? Or should I have written it as the following,
$$
\frac{\partial f(x, y)}{\partial x} = \frac{\partial f(u, y)}{\partial u} \cdot \frac{d u(x)}{dx}
$$
Or are both the above expressions correct?
 A: 
Is the above expression correct? Or should I have written it as the following,
$$
\frac{\partial f}{\partial x} = \frac{\partial f}{\partial u} \cdot \color{blue}{\frac{d u}{dx}}
$$

As $u$ is a function of one variable only, this (in blue) would indeed be the standard notation.

$$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial u} \cdot \color{red}{\frac{\partial u}{\partial x}}$$

The above notation would probably be interpreted correctly, but I would advise against using it to avoid confusion.
See for example: Mathworld: Chain Rule.
A: Both forms are correct, but that with partial derivative is very strange to see.
Why do you want to use partial derivative symbol with a single variable function?
The partial derivative makes sense with $f$ because it means the (total) derivative of $f$ restricted to the straight line parallel to the $x$ axis (thereby becoming a one variable function on which a (total) derivative can be applied). For $u$ you do not need to restrict it to anything so no special symbol different from the ordinary derivative symbol must be used.
