Question

Please help!! I don't know where my mistake is

Answer 1

\[x^3+y^3=18xy\] is that the start?

Answer 2

Yup!

Answer 3

and you need to find \(y'\) by implicit diff right?

Answer 4

Yeah.

Answer 5

the mistake was in the first line
when you take the derivative on the right, you get
\[18(xy'+x)\] like you wrote,
but on the left you should get
\[3x^2+3y^2y'\]

Answer 6

Oh. okay. Why is there a y prime there though?

Answer 7

lets go slow

Answer 8

you have an equation \[x^3+y^3=18xy\] which is some curve and may not represent a function (may not pass the vertical line test) but it does locally
so you are thinking that \(y\) is a function of \(x\) even though you do not know it explicitly (hence implicit diff)
so lets call \(y=f(x)\) and rewrite this as
\[x^3+f^3(x)=18xf(x)\]

Answer 9

Okay.

Answer 10

when you take the derivative on the right, you need the product rule
\[18f(x)+18xf'(x)\] more easily written as
\[18(y+xy')\] as you wrote

Answer 11

on the left, you need the chain rule
you get
\[3x^2+3f^2(x)f'(x)\] more easily written as
\[3x^2+3y^2y'\]

Answer 12

just like if you had
\[x^3+\sin^3(x)\] the derivative would be \[3x^2+3\sin^2(x)\cos(x)\]

Answer 13

Okay. I was a little confused, but the last post cleared up my confusion.

Answer 14

hope so
then it is algebra from there on in to isolate \(y'\) once you have
\[3x^2+3y^2y'=18xy'+18y\]

Answer 15

Okay. So when I solve problems with implicit diff, will there always be a y prime if I am finding the derivative of a term that has a y variable in it?