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There is some sense in which the derivative of a function $\frac{df}{dx}$ can be written as a "product" $Df$. And while solving, treat $D$ as a "number".

What is this process called, if it even has a name?

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  • $\begingroup$ You're examples are wildly different, but you give them as example of the same thing. Very unclear question. $\endgroup$ Mar 30, 2021 at 15:44
  • $\begingroup$ @JensRenders In what sense are they different, if you don't mind me asking? Is it because $exp$ is a function, while $\frac{d}{dx}$ is an operator? $\endgroup$
    – Sofviic
    Mar 30, 2021 at 15:46
  • $\begingroup$ regarding the application of a linear operator as product has nothing to do with defining a shorthand exp for the function $e^x$ $\endgroup$ Mar 30, 2021 at 15:50
  • $\begingroup$ @JensRenders I see, I'll remove the mention of $exp$ then. $\endgroup$
    – Sofviic
    Mar 30, 2021 at 15:55
  • $\begingroup$ I don't think this has a name though $\endgroup$ Mar 30, 2021 at 16:01

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There's a very general term "[by] abuse of notation", for when you treat objects of type X as if they follow the rules for objects of type Y, without having established that they obey those rules. An example might be writing an infinite series of terms without having defined convergence.

And as noted by the OP, one could also describe this as "[by] analogy with Y".

Whichever description one uses, you've admitted that you've left the path of rigorous deduction, and what follow may not be true/valid. Usually one follows up with either

1 - Going back and rigorously proving that X's do behave as you presumed, or

2 - If all you wanted was to find a solution to some problem, you prove that the answer you found does in fact provide a solution, and the way that you found it is considered to just be a trick or heuristic or ansatz.

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