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If we are asked to find the derivative of x³ with respect to x, the answer will be 3x² (y=x³, dy/dx=3x²).

So dy=3x²*dx, I understood this till here, but in the next step my teacher told us that we can write this as: d(x³) which is fine but how do we graphically understand this ?

For example dy/dx is basically an infinitesimally small change in y with respect to x which can be graphically understood by slope of the tangent at that particular point in the curve.

So in a similar fashion, how do we understand d(x³) ?

Thank you.

Wallflower06
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    In short, don't worry about the name if it's referring to the same object. y and x^3 are the same thing, given that they are equal to each other. So their meaning under the differential operator is the same. – Joseph_Kopp Aug 26 '23 at 16:06

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There are a few things going on here, so I will try to explain.

Formally, the derivative is an operator: it takes in a function and outputs a different function, which as you correctly pointed out represents the slope of the tangent when evaluated at a particular point. That means I can write it as $\frac{d}{dx}(\cdot)$ where if you give me a function $f$, I will give you back the function $\frac{d}{dx}(f)$.

If you think of it this way, if you pass in the same input you will get back the same output and since $y = x^3$, $\frac{d}{dx}(y) = \frac{d}{dx}(x^3)$. We often write the function whose derivative we are taking on top, so that it corresponds with the "change in y over change in x" intuition we have about its value at any given point.

Now, note that in the above I'm using $\frac{d}{dx}$ as a monobloc of an expression. That means I'm not actually thinking of it as a fraction. This brings me to something you wrote, which is $dy=3x^2 dx$. If we say that $dy$ and $dx$ represent the change in y and x respectively, then the above expression is perfectly fine.

However, we have to be careful, because $dy$ and $dx$ are, as you said, infinitesimal quantities. When you learn about integration and specifically change of variables, you will be multiplying and dividing by these infinitesimals all the time. But can we really multiply and divide by them as we would otherwise? Moreover, how does this fit in with the definition of the derivative as an operator? If $\frac{d}{dx}$ is an operator, do $dy$, $dx^3$, dx, have any standalone meaning?

All of these will come together when you learn about differential forms, much later down the road, if you choose to study mathematics. For now, it will probably serve you best to juggle between the different notations and intuitions of the derivative.

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