Which way is correct to represent this differentiation? $$3x^2+x^3$$
Option 1
$$\frac{d}{dx}(3x^2+x^3)=6x+3x^2$$
Option 2
$$\frac{d(3x^2+x^3)}{dx}=6x+3x^2$$
My student wrote the Option 2 style, not sure whether I can consider is ok or not.
 A: They are both correct, but represent slightly different ways of thinking about the derivative (at least in my mind).
When I read option 1, what I think is that you have this "machine" called "the derivative" which is written $\frac d{dx}$. You feed functions to it, and it gives you a (potentially) different function back which is related to the function you gave it by taking the relevant limits, or more intuitively by finding the "steepness" of $f$ at all points.
As for option 2, it makes me think that for any function $f$, you have an associated function called "the derivative of $f$" which is written $\frac{df}{dx}$ (even though it looks like it consists of four letters and a line, it's really just one big symbol, kind of like how the letter 'i' is just one symbol even though it consists of both a line and a dot). It is defined by the relevant limit-taking or steepness-interpretation.
A: Both of the given representations are equivalent.
For:
$$y(x)=3x^2+x^3$$
$$\frac{dy}{dx} \equiv \frac{d(3x^2+x^3)}{dx} \equiv \frac{d}{dx}(3x^2+x^3) \equiv  \frac{d(3x^2)}{dx}+\frac{d(x^3)}{dx} \equiv \frac{d}{dx}(3x^2)+\frac{d}{dx}(x^3) 
=6x+3x^2$$
