Question

I just want somebody to check my work on this (Multivariable Calc implicit differentiation problem.)

Answer 1

Hmm I can't read your handwriting :c
is that a 4 or an x on the right side of the equation?\[\Large\rm x^2+y^2+z^2=\color{orangered}{4}yz\]

Answer 2

Lol, I always think your icon is a picture of a pizza when I see it. XD

Answer 3

Yep, that's a 4.

Answer 4

Ugh I must be a little rusty with partials :c
this question doesn't make sense to me.

Like if you take the partial of each side with respect to z,
then y is constant as well.\[\Large\rm \frac{\partial}{\partial z}\left(x^2+y^2+z^2\right)=\frac{\partial}{\partial z}4yz\]Giving us,\[\Large\rm 0+0+2z=4y\]
Hmmm maybe I need to brush up on my partials >.<

Answer 5

Let me put up the actual question prompt to see if I'm misinterpreting it at all. One moment.

Answer 6

I'm trying to avoid using the Implicit Function Theorem if possible, but whatever works at the end of the day, works.

Answer 7

Mmmm ok I think I get it now.
Lemme look back at your work a sec.

Answer 8

I would divide both sides by 2 to simplify things before going any further than that step.

Answer 9

Your answer looks very close though.

Answer 10

I guess I was being silly...
I forgot that we were thinking of y as a function of the other two variables.\[\Large\rm x^2+y(x,z)^2+z^2=4y(x,z)z\]

So differentiating gives:\[\Large\rm 0+2y\frac{\partial y}{\partial z}+2z=4\left(\frac{\partial y}{\partial z}z+y\right)\]Looks like you did your product rule correctly, that's good.

Answer 11

Whoah, didn't see any notifications, sorry, lemme take a look.

Answer 12

Alright, cool! Thanks so much. Other than the tiny carrying over the four that I forgot, I guess everything else looks good?

Answer 13

In your very last step ( which I erased in my picture ),
you did the division backwards.

Your numerator and denominator are backwards! :O
Careful there!

Answer 14

Looks like you've got the process pretty well figured out.
Just gotta watch the arithmetic :)