Please for the love of god help me. I have a fundamental misunderstanding of differentials and Leibniz notation. I'm confused as to even where I should begin. Please allow me to start off my explanation by showing how my book introduces u-substition: I have highlighted in red the parts I am confused on. Let me start... Why are we dealing with differentials!? Aren't differentials supposed to measure the "change in linearization"?! Differentials were taught to me in the context of approximations, for the life of me I cannot see the connection with integrals. Where do differentials and approximations and errors come into play here? Why am I dealing with differentials? On top of all of this I still do not understand why dy/dx = f'(x) can be written as dy = f'(x)dx (since I'm told dy/dx is not actually a fraction), which is just adding to my confusion. I am told that Leibniz was wrong on some things and that we do not take the dx stuff seriously and that I should ignore it, but it's right in this context??? Or it's sort of right in a way but not really? And how can you just interpret it AS IF it were a differential? What does that actually mean? Interpret "as if"?!?! This doesn't sound like math, it sounds like fancy handwaving that somehow produces a correct answer through magic. How do they just "interpret" something as something else? I am so lost that I'm about ready to fall out of my chair. Please somebody help. Spare no detail... I have searched Google until the early hours of the morning for weeks like a madman trying to figure this out.