Major notation doubt in calculus.

Question

Which of the following notations is correct?

$$\frac{d}{dx}(y)$$ or $$\frac{dy}{dx}.$$

Please don't think this is trivial.

Answer 1

They are both correct, and they mean the same thing. If there is any difference, it's in the mind set they convey.

$\frac{dy}{dx}$ is a function defined as the derivative of $y$. It's a single symbol. On the other hand, $\frac d{dx}y$ is the result of applying the differentiation operator to the function $y$. It's two symbols (one function, one operator).

Answer 2

Both the notations are correct. Both of them mean the derivative of $y$ w.r.t $x$.

Answer 3

Here some background:

On one hand, you can see the differentiation as a linear operator acting on functions. So, you may see $\frac{d}{dx}$ as a mapping that assigns to a (differentiable) function its derivative. Hence, you may write $$\frac{d}{dx}y$$

Another way to see it, is to interpret the limit of the difference quotient as a quotient of "infinitesimally" small differences, which are called differentials. So, you may write $$\lim_{\Delta x \rightarrow 0}\frac{\Delta y}{\Delta x} = \frac{dy}{dx}$$