Question

Find d2y/dx2 by implicit differentiation, given x^2 + 3y^2 = 5. Help please!

Answer 1

step number 1 take the derivative wrt x you get
\[2x+6yy'=0\] is that part ok?

Answer 2

yes I got that

Answer 3

then solve for
\[y'\] via
\[6yy'=-2x\]
\[y'=-\frac{x}{3y}\]

Answer 4

Mhm I got that far

Answer 5

so far so good?

Answer 6

Yes :)

Answer 7

ok so now we do it again, this time with the paininthebutt quotient rule

Answer 8

OOOOH

Answer 9

want to factor out the
\[-\frac{1}{3}\] first or leave it in?

Answer 10

we can leave it in I guess. Which ever one is easier...

Answer 11

lets make life easy and factor it out so we have
\[-\frac{1}{3}\times \frac{x}{y}\] and we can mutiply by
\[-\frac{1}{3}\] at the end

Answer 12

ok :)

Answer 13

so now for
\[y''=-\frac{1}{3}\times \frac{y-xy'}{y^2}\]

Answer 14

we are not done yet, but is that ok?

Answer 15

yes :)

Answer 16

ok and last step is an algebra one. because we actually know what
\[y'\] is

Answer 17

\[y'=-\frac{x}{3y}\]so that is what we replace it by in our second derivative to get
\[y''=-\frac{1}{3}\times \frac{y-x\times -\frac{x}{3y}}{y^2}\]

Answer 18

still not done. more algebra

Answer 19

ok >.<

Answer 20

need to clear the compound fraction, so maybe multply top and bottom by
\[3y\] to get
\[y''=-\frac{3y^2+x^2}{9y^3}\]

Answer 21

and we are STILL NOT DONE

Answer 22

because in a fit of trickery they have arranged it so that you numerator is in fact what you started with (be on the lookout for this)
recall that we started with
\[x^2+3y^2=5\] so the numerator is just...
5! and you get
\[y''=-\frac{5}{9y^3}\] (check my algebra)

Answer 23

You... are a genius. I was about to cry. Thank you sooo much!

Answer 24

actually myininay will check my algebra, she loves finding my mistakes

Answer 25

ooops i got a medal. i guess it is right

Answer 26

You were completely correct! Thank you again!

Answer 27

well there is one thing i don't like that you said...

Answer 28

yw

Answer 29

i just don't like \[\times \] this symbol

Answer 30

but goof job

Answer 31

i mean good job lol

Answer 32

oh i thought you were going to tell me i was off by a minus sign or something. i put that in to make sure it is clear that we were multiplying.
i like "goof job" better actually

Answer 33

me too