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I was relatively comfortable with Leibniz's notation of dy, dx as representing infinitesimally small values of change in y and x, however I was watching a Math lecture yesterday where I saw something I was simply unable to conceptualise.

I've always seen the d as an operator, that functions in the same way as the sine, cosine, tan, log, etc functions. d(x) is taking the input x and performing the differential operation on it, outputting .dx -> which is the infinitesimally small change in the variable x. This by itself is meaningless, unless it is put in a statement with another infinitesimal, such as dy, whereby ratios give meaning to the expression.

However, I saw the use of the annihilation operator in the following expression:

(d/dx + x) y = dy/dx + xy

I know algebraically it works, but it treats d as if it is some simple constant. Seeing d/dx by itself was too difficult to conceptualise, I can't grasp the meaning of it. For me it was akin to seeing:

(sin( /sin(x) + x)y = sin(y)/sin(x) + xy.

So, my questions are:

1. Can you simply pull the input of a differential away from the expression

-> dx + x = (d + 1)x

2. How do you conceptualise what the meaning of d/dx is. What does it actually mean?

Thanks for your help!

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# Conceptualisation of d/dx

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