u/window2020

The problem was (x+1/x)^2

I solved it: = (x+x^-1)^2

using udv +vdu
=2(x+x^-1) (1-x^-2)
=(2)(x+1/x)(1-1/x^2)
This is the correct answer per my book.

Here’s my question: If I multiply it out and take the derivative I get a different answer:
(x+x^-1)^2 = (x+x^-1) (x+x^-1) = X^2 + 2x^0 + x^-2

= (2x) (1-x^-2)

I can’t seem to get it in to the same form as the given correct answer. Is it my algebra, or is it not a valid approach to multiply it out first and take the derivative? Thanks

Someone I know who owns a Tesla told me that there is some kind of warming the battery process before charging that he is supposed to use. I don’t know if this is to increase the rate of charging or the efficiency of charging. What is this all about?

I also remember from taking physics (a long time ago) that a battery, like a old-fashioned lead acid car battery would generate less power (I don’t know if it’s voltage or amperage that I’m talking about) if the battery is cold, then if it is warm. But the rate of spontaneous discharge of the battery is lower and a cold environment. Is what I’m saying accurate?

I am trying to teach myself calculus. This is a problem from my book. I did it multiple times and finally got it right according to the answer in the book: 12x^5 + 25x^4 - 9x^2 - 2x - 2

The problem is, I checked it on chatGPT and it gives 10x^5 + 20x^4 + 4x^3 - 6x^2 - 2x - 2

Then I checked it on Gemini and it gives 12x^5 + 25x^4 - 11x^2 - 2

So I am going nuts. Which is correct? Please, please check my work. Thank you