Question

Use chain rule to find dy/dx for the given value of x.

y+ (u-1/u+1)^1/2

u=sqrt x-1; for x=34/9

Answer 1

first step I get
1/2 (u-1/u+1)^-1/2
not sure what to do next. I am brain dead at the moment.

Answer 2

what was original function
did you replace dx with du
du = dx/2sqrt(x-1)

Answer 3

sorry i think im mistaken, ignore what i just said
but you will use du

Answer 4

take what you have and you will multiply it by derivative of (u-1)/(u+1)
use quotient rule

Answer 5

what do I do with the ^1/2. I need to bring that infront of the equation and subtract 1 from the 1/2, is that correct?

Answer 6

correct looks like you did that already in the 1st post

Answer 7

ok, after that do I use quotient rule?

Answer 8

yes
(f'g -fg')/g^2

Answer 9

ok after that I get (u+1)-(u-1)/(u+1)^2 which ends up being 2/(u+1)^2

Answer 10

correct, now simplify and then you have to multiply by du ans replace u with x

Answer 11

do I need to take derivative of U first?

Answer 12

yes
sorry that was what i meant by du (derivative of u)
then replace all the u's with sqrt(x-1)

Answer 13

so derivative of u would be 1/2(x-1)^-1/2

Answer 14

correct

Answer 15

do I replace the u with 1/2(x-1)^-1/2 and also replace the x with 34/9 or will that be a different step?

Answer 16

no remember what you assigned u as
u=sqrt(x-1)
1/2(x-1)^-1/2 just gets multiplied on the outside
Last step will be replacing x with 34/9

Answer 17

so
2/(u+1)^2 * 1/2(x-1)^-1/2

U= sqrt x-1 and x=34/9
????

Answer 18

yes but dont forget what you had originally
1/2 (u-1/u+1)^-1/2
thats part of it still too

Answer 19

I am lost now

would it be

1/2 [(sqrt x-1)-1]/sqrt [(x-1)+1]^-1/2 * 1/2 (34/9-1)^-1/2
????

Answer 20

2/(u+1)^2
correct just add this part
with u = sqrt(x-1)
2/[sqrt(x-1)+1]^2
also replace all the x's with 34/9
i know this one has a lot of parts

Answer 21

this answer really seems wrong, I am sure I did something wrong in my math but I get 2025/136

Answer 22

sorry wrong

2025/128

Answer 23

hmm i worked it out and got 27/320
ill write it out so you can compare
\[\frac{dy}{dx} = \frac{1}{(2\sqrt{(34/9)-1})(\sqrt{(34/9)-1}+1)^{2}\sqrt{\frac{\sqrt{(34/9)-1}-1}{\sqrt{(34/9)-1}+1}}}\]
34/9 -1 =25/9
sqrt of that is 5/3

Answer 24

why did you take 1 over everything

Answer 25

because everything had a power of -1/2 right, that means its on the denominator

Answer 26

oh I see now, didn't think of that

Answer 27

so what equation I am using to replace with?

Answer 28

??

Answer 29

it is hard for me to explain to you what I am talking about without showing you my paper or pointing to what you have written.

I can see where the (sqrt 34/9)-1-1/(sqrt 34/9)-1+1 came from, the deriviative of original problem 1/2(u-1/u+1)^-1/2.
But not the (2sqrt 34/9)-1) (sqrt 34/9 -1+1)^2

Answer 30

ahh it comes from using the chain rule, we are multiplying derivative of original times derivative of inside function
ex:
f(x) = sqrt(x^2 +1)
its a composite function, i say g(x) = x^2+1
f(g) = sqrt(g)
f'(g) = 1/2sqrt(g)
g'(x) = 2x
whats important is the derivative of original is product of these 2 derivatives
f'(x) = g'(x)*f'(g)
f'(x) = 2x*1/2sqrt(x^2+1)
so at the end we replace x^2 +1 back in for g