okay so just out of curiousity

if i had

x² + xy + y² = 1

how would i differentiate?

Question

okay so just out of curiousity


if i had

x² + xy + y² = 1

how would i differentiate?

hartnn · Answer

heard of implicit differentiation ?

amorfide · Answer

no i have never heard of it

kainui · Answer

you really just use the chain rule. Most of the time the derivative with respect to x is 1, so you don't think about the derivative of 3x^2 being 3x*1, but when you do implicit differentiation you take the derivative of 3y^2 to be 3y*(dy/dx)

hartnn · Answer

u want to diff w..r.t x, right ?

kainui · Answer

6* not 3x or 3y by the way! lol

amorfide · Answer

yeah rtx

hartnn · Answer

ok, so u know product rule, and chain rule,
can u differentiate xy ?

amorfide · Answer

no i can not lol
i have never had xy together only y=x or something

hartnn · Answer

but u know product rule right ? 
take x and y as different functions and diff. xy using product rule

hartnn · Answer

like x'y+xy'
and x' = ?

amorfide · Answer

would it be y+x?

hartnn · Answer

umm.....x'=1 right ?
so diff. of xy  will be y+xy'
ok

amorfide · Answer

but isnt y' also 1? so it would be y+x?

hartnn · Answer

y' would have been 1 if you would have diff. w.r.t y
as in d/dy of y =1
but here you are differentiating w.r.t x
do its diff. would be 
dy/dx that is y'
makes sense ?

amorfide · Answer

ohhh okay! thank you, that makes more sense

hartnn · Answer

and what will be diff.  of y^2 then ?

amorfide · Answer

2y²' ?

hartnn · Answer

since d/dx of x^2 = 2x
d/dy of y^2 = 2y dy/dx = 2yy' 
ok?

amorfide · Answer

okay so the diff of y^3 would be 3y²y' ?

hartnn · Answer

yup

amorfide · Answer

x² + xy + y² = 1

soo
2x+(x+y')+2y'=0 is that right

hartnn · Answer

2nd term ? 
 y+xy'

amorfide · Answer

oops yeah that is what i meant

2x+(y+xy')+2y'=0

amorfide · Answer

2x+(y+xy')+2yy' =0

hartnn · Answer

yes, now find y' from there ?

amorfide · Answer

how do i do that? lol

amorfide · Answer

2x+(y+xy')+2yy' =0

-2x=y+xy' +2yy'

-2x-y=xy'+2yy'

-2x-y/2y=xy'/2y   +y'

uhm im stuck

hartnn · Answer

2x+(y+xy')+2yy' =0
xy'+2yy' = -2x-y
(x+2y)y' = -(2x+y)
now ?

amorfide · Answer

(x+2y)y' = -(2x+y)

y'=-(2x+y)/(x+2y)

hartnn · Answer

yes :)

hartnn · Answer

wasn't it easy ?

amorfide · Answer

thank you! is that the entire thing finished?

hartnn · Answer

yup., u got your y'

amorfide · Answer

easier than i thought, thank you, i was just curious how to do a question like this, it isn't needed :P
you are awesome!

hartnn · Answer

welcome ^_^