Notation question.  Where does the x in the denominator come from?

The last class I had in math was college precalculus.  I'm a bit older now.....70.
I'm reading Calculus Made Easy. This is from Chapter IV, Simplest Cases.  I suspect I'm not remembering or didn't learn a mathematical rule or how it should be applied in calculus.  I've looked everywhere I can think of and can't find the answer.

y+dy=(x+dx)^-2
           
                =x^-2(1+dx/x)^-2

(it goes on to expand by using the binomial theorem which I have no question about right now)

In the last part of the equation inside the parentheses, where/how does the x in the denominator of dx/x come from?  Mathway leaves out both the x in the numerator and denominator leaving d by itself. It also puts the whole equation in the denominator (I get that part even though it is not used here).  My understanding is that d is not equal to dx which is not a variable?

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Kav10 Kav10
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  • I guess I need the steps between equation 1 and 2. How did the x in the denominator get there? I'm trying to go backwards, but.... You pull the x out to get 1, then what?

    • Kav10 Kav10
      0

      You just take the x out. That's it. You asked how equation 1 got to equation 3, so I added equation 2, which shows exactly it is done. x is factored out.

  • Or, said another way, how does dx=dx/x?

    • Kav10 Kav10
      0

      dx does not equal dx/x. x+dx equals x(1+dx/x).

  • Ok, I think I get it. I'm not up on my factoring either. I'll get there. I'll return your coffee too. :)

    • Kav10 Kav10
      0

      Sounds good. Let me know if it still does not make sense. If it does, please accept the answer.

  • I found this exact same question on Stack Exchange just now which I find humorous. I feel somewhat vindicated of my confusion. :) https://math.stackexchange.com/questions/4622759/factoring-x-out-of-xdx Thank you very much.

    • Kav10 Kav10
      0

      That's interesting! I am glad that you got it now.

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