was substituted for the equivalent expression of which is why you used your sub.
It's the same principle of given that
I understand the basics of integration, but I'm extremely rusty, and in the process of refreshing myself, I hit an example I don't understand.
This is the problem:
So, my understanding of the correct way to solve this is this:
Here's what I don't understand - what happened to ? ?
To me, it just dissappeared.
Just in case a picture helps to follow your way back through the chain rule...
... where (key in spoiler) ...
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!
That helped!. I think I'm starting to grasp it now. Part of what I didn't understand is that if is coming from differentiating , where did the original go. But I think I understand that that was being swapped for the due to the equality, rather than being inserted.
It also explains the second part of what I didn't understand (but didn't post) which is why, if the numerator was did the answer equal this one, but with a as a multiplier in front of everything.
I was using Wolfram Alpha to show me the steps, but the problem was that it wasn't showing me enough of the intermediate steps for me to fully grasp it.
Thank you! And thanks to everyone else who tried to knock this into my brain as well.