Please for the love of god help me.(adsbygoogle = window.adsbygoogle || []).push({});

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"?! (1) [tex]\int 2x\sqrt{1+x^2}dx[/tex]

To find this integral we use the problem-solving strategy ofintroducing something extra. Here "something extra" is a new variable; we change from the variablexto a new variableu. Suppose we letube the quantity under the root sign in (1), then theDIFFERENTIALofuisdu = 2xdx. Notice that if thedxin the notation for an integral were to beINTERPRETED AS A DIFFERENTIAL, then the differential2xdxwould occur in (1) and, so, formally, without justifying our calculation, we could write

(2) [tex]\int 2x\sqrt{1+x^2}dx = \int \sqrt{u}du = \frac{2}{3}u^\frac{3}{2}+C=\frac{2}{3}(x^2+1)^\frac{3}{2} + C[/tex]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 justinterpret it AS IFit 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.

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# Differentials - I'm HOPELESSLY Confused. PLEASE Help. PLEASE

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