Question

Given xy+y^2=x+2, find d^2y/dx^2

Answer 1

first implicit derivative:
1 * y + x * y ' + 2*y * y ' = 1

Answer 2

this is implicit differentiation
treat y as a function of x
use the product rule/chain rule for xy and chain rule for y^2:-

(x *dy/dx + y*1) + 2y * dy/dx = 1

now make dy/dx the subject

perform similar process to find the second derivative

Answer 3

I'm tryinggggg :/ I found dy/dx=(1-y)/(x+2y) right?

Answer 4

yes

Answer 5

now apply the quotient rule

Answer 6

I first did it in terms of dy/dx & then substituted dy/dx by (1-y)/(x+2y).
Now, a quick question.
If you have this:
(1-y)((x+2)/(x+2y))
What do you do exactly?
You distribute the (1-y) to the numerator only?

Answer 7

it will be easy if u dont take the term to denominator

Answer 8

1*y+x*dy/dx+2y*dy/dx=1

Answer 9

now first differentiate this but dont move any term on right

Answer 10

differentiation of dy/dx=d^2/dx^2

Answer 11

oh - I see - good way to do it!

Answer 12

thanks bro :)

Answer 13

I don't think I get what you mean :p
=> dy/dx(x+2y)+y=1 ??

Answer 14

now you differentiate that implicitly
as gorv said derivative of dy/dx is d^2y/dx^2

Answer 15

so you use the product rule to differentiate dy/dx(x + 2y)

Answer 16

so we have
dy/dx ( 1 + 2 * dy/dx)) + d^2y/dx^2( x + 2y) + dy/dx = 0

Answer 17

now make d^2y/ dx^2 the subject the substitute for dy/dx

Answer 18

It sounds easier, but sorry I'm not getting it :/ I did it your way (which is also the way i would've done it) & it's pretty long :p I just got lost in the simplifications

Answer 19

yes - its a bit messy my way but theres no reason why it can't be done

Answer 20

Actually, wait

Answer 21

to be honest i havent tried it my way - i'll look at it now

Answer 22

have you understood gorv's method?

Answer 23

Brb

Answer 24

sorry i gotta go - i'll look at again later

Answer 25

must go

Answer 26

Ahhhh, yesssssss!! I got it :DD Woahh, much easier! THANK YOUUU

Answer 27

No need to look at it again! Thankssss! Bye :)

Answer 28

bye