Question

guys ,,, can u answer this question
double differentiate
xy-y^2=10

Answer 1

its not so much an answer as it is a process

Answer 2

and i assume you mean you want to derive it twice?

Answer 3

yes

Answer 4

do you know how to derive? i mean the rules such as power rule, product rule, chain rule etc?

Answer 5

i think i should use product rule

Answer 6

why do u always say derive amistre lol

Answer 7

this takes both a product rule and a power rule in the first derivative

Answer 8

:) its how i make sense of the sale on Dwords that calculus bought

Answer 9

lol

Answer 10

d (xy-y^2=10)
---
dx


d (xy) d (-y^2) d (10)
--- + ---- = -----
dx dx dx

Answer 11

first i got -y/(x+2y)

Answer 12

(xy)'=xy'+y
(-y^2)'=-2yy'

Answer 13

(constant)'=0

Answer 14

dx dy dx
--- (y) + --- (-2y) = -- (0) ; and dx/dx = 1
dx dx dx

Answer 15

.... yeah, i messed up that xy, forgot to do a product rule :)

Answer 16

dx dy dy dx
--- (y) + --- (x) + --- (-2y) = -- (0) ; and dx/dx = 1
dx dx dx dx

thats better

Answer 17

dy dy
y + --- (x) + --- (-2y) = 0
dx dx

Answer 18

it will be dy/dx=-y/x+2y

Answer 19

right

Answer 20

derive it again i spose


d (y + y'x + y' -2y = 0)
---
dx


d (y) d (xy') d (-2y y') d (0)
--- + --- + --- = ---
dx dx dx dx



dy dx d2y d2y dy dx
--- (1) + --- (y') +----(x) + ---- (-2y) + --- (-2y') = --- (0)
dx dx dx2 dx2 dx dx

Answer 21

i would had solved for y' in the first one
but lets see what happens with amistre;s approach

Answer 22

y' + y' + y'' x +y'' -2y + y' -2y' = 0

2y'+ y''(x -2y) -2(y'^2) = 0

y''(x -2y) = 2(y'^2) - 2y'

y'' = 2y'(y'-1)/(x-2y)

Answer 23

ok

Answer 24

since:

y' = -y/(x-2y) we get...

\[y'' = \frac{-2y}{(x-2y)}*\frac{\frac{-y}{(x-2y)} -1}{(x-2y)}\]

\[y'' = \frac{-2y}{(x-2y)}*\frac{\frac{-y-(x-2y)}{(x-2y)}}{(x-2y)}\]

\[y'' = \frac{-2y}{(x-2y)}*\frac{-y-x+2y}{(x-2y)^2}\]

\[y'' = \frac{-2y}{(x-2y)}*\frac{y-x}{(x-2y)^2}\]

\[y'' = \frac{-2y^2+2xy}{(x-2y)^3}\]
perhaps?

Answer 25

prolly messed it up along the way :)

Answer 26

i think it looks good

Answer 27

then I do too :)

Answer 28

:)

Answer 29

wait i dont know

Answer 30

\[xy-y^2=10\]
\[xy'+y-2yy'=0\]
\[y'(x-2y)=-y\]
\[y'=\frac{-y}{x-2y}\]
\[y''=\frac{-y'(x-2y)-(1-2y')(-y)}{(x-2y)^2}\]
\[y''=\frac{\frac{y}{x-2y}(x-2y)-(1+\frac{2y}{x-2y})(-y)}{(x-2y)^2}\]
\[y''=\frac{y+y+\frac{2y^2}{x-2y}}{(x-2y)^2}\]
\[y''=\frac{2y(x-2y)+2y^2}{(x-2y)^3}\]
\[y''=\frac{2xy-2y^2+2y^2}{(x-2y)^3}\]
\[y''=\frac{2xy}{(x-2y)^3}\]
maybe i made a mistake though

Answer 31

oops i did you did good amistre

Answer 32

see 2y(x-2y)=2xy-4y^2
and i put 2xy-2y^2

Answer 33

\[y''=\frac{2xy-4y^2+2y^2}{(x-2y)^3}=\frac{2xy-2y^2}{(x-2y)^3}\]

Answer 34

yeah, i missed it on the algebra lol

Answer 35

no you did good
and i did bad
or do you mean you didn't see my algebraic mistake?

Answer 36

i was going thru your work thinking it was mine :)

Answer 37

but i see now we got the same results :)

Answer 38

we apparently have the same handwriting

Answer 39

lol

Answer 40

it looks similar

Answer 41

we do have different writing styles though
you use dy/dx notation
and i use y'

Answer 42

i like to use d/dx to show the background movements of implicit differentation

Answer 43

people seem to have a notion that it is some magical conflageration that is somehow different from what they learn derivatives to be

Answer 44

i like y' because i don't like writing fractions
it is just easier for me to write y' then to write dy/dx

Answer 45

y' cleans it up nice for me

Answer 46

either notation is fine
but i think dy/dx notation is suppose to be better
and i cant remember why

Answer 47

pros and cons, thats all

Answer 48

i think they said Europe or some country/continent was behind in math because they never used dy/dx notation
i;m sure but im just recalling false stuff

Answer 49

maybe im recalling false stuff*

Answer 50

thats right; england stuck to newtons notation (y') while the rest of the europe used leibniz (dy/dx) and england fell behind

Answer 51

right
y' is newton's
and
dy/dx was leibniz
you're so smart
:)

Answer 52

i would had fell behind with england then

Answer 53

amistre=the rest of europe

Answer 54

us sicilians never worried to much about it ;)

Answer 55

i want their pizza