Question

The derivative of y=(x+x^-1)^2 and why I can't use chain rule?

Answer 1

why do you think you can use chain rule ?
you can.

Answer 2

oh. well, when I tried to use it, I got a wrong answer, a really complicated one...

Answer 3

then try expanding using \((a+b)^2=a^2+2ab+b^2\)
this is a simpler way.

Answer 4

a=x,b=x^-1

Answer 5

really? hmmm I had my dad do the problem, and he did the chain rule, except without the du/dx, and he couldn't relly explain why... it seemed simpler, but I was confused on why he didn't put the du/dx~

Answer 6

if chain rule is used, we need du/dx
anyways, did you try expanding (x+1/x)^2 ?

Answer 7

x^-1 times x^-1 is adding the exponents? verdad?

Answer 8

\(x^{-1} \times x^{-1}=x^{-1-1}=..?\)

Answer 9

okay, cool. I got 2(1-x^-3) when I expanded and took the derivative.

Answer 10

umm..how ?
before taking the derivative, did u get
\(x^2+2+x^{-2}\)
??

Answer 11

OH. whoops. 2x...? lol okay, I'll try it again.

Answer 12

2(1-x^-3) is almost correct, just one term is in error

Answer 13

when you factor out an x from something like (2x-2x^-3), what would you get for the second term in the paretheses?

Answer 14

(2x-2x^-3), is correct answer, you can factor out 2 to get
2(x-x^-3)
if you want to factor out x also, (which is not necessary, but since you've asked)
\(2(x-x^{-3})=2(x-x \times x^{-4})=2x(1-x^{-4})\)

Answer 15

okay, YES. wait how would you do this problem with the chain rule? I think I'll still need that for the test tomrorw...

Answer 16

y= u^2
u =x+1/x
dy/dx=...?

Answer 17

2(x+1/x)=dy/dx?

Answer 18

what ?
you know the definition of chain rule, right ?

Answer 19

\([f(g(x))]' = f'(g(x))g'(x)\)

Answer 20

y=u^2
dy/dx = (u^2)' du/dx

Answer 21

and du/dx=..?

Answer 22

oh, you just did 2u and got 2(x+1/x) right ? you need to multiply it bu du/dx also

Answer 23

so I need to multiply that by du/dx to get dy/dx???

Answer 24

yes because,
dy/dx = (dy/du) * (du/dx)

Answer 25

So then idy/dx is basically like finding the derivative of the entire function and du/dx is like finding the derivative of only the inside?

Answer 26

thats correct :)

Answer 27

So then that sepcific question that we were talking about earlier would become 2(x+1/x)(1+-1x^-2)?

Answer 28

that would be correct :) good.

Answer 29

if you simplify that, you would again gget (2x-2x^-3)

Answer 30

I ended up at 2x+4/x+2/x^3...?

Answer 31

you tried to simplify this:
2(x+1/x)(1-1x^-2)
right ?
how come there's no - *minus* sign :P

Answer 32

OH!!! let me try again~ :)

Answer 33

sure, take your time , but do it correctly this time :)

Answer 34

2x-2/x^3?

Answer 35

that.is.correct.:)

Answer 36

which is same what we got...

Answer 37

oh yeah~:) phfft I was totally thinking when I pu that post up...lol
thank you!!!here, best response!!! <3 cya

Answer 38

welcome ^_^
cya :)