Implicit derivative difficulties

[kald13]

Homework Statement

I am struggling with implicit derivatives, and though my course book includes final solutions to odd numbered exercises, it does not show the work. As such, I'm stuck in the process of getting from point A to point B:

Find the derivative y'(x) implicitly of

$((x+3)/y)=4x+y^2$

2. The attempt at a solution

I know to start with the derivative of each side of the equation:

$((d/dx)[x+3]y-(x+3)(d/dx)[y])/y^2=(d/dx)[4x+y^2]$
$((1)y-(x+3)(y'))/y^2=4+2y(y')$

Multiply both sides by $y^2$

$y-y'(x+3)=4y^2+2y^3(y')$

Subtract $y$ and then divide both sides by $-(x+3)$

$-y'(x+3)=4y^2-y+2y^3(y')$
$y'=-(4y^2-y+2y^3(y'))/(x+3)$

So now I have y' isolated on the left, but I still have a y' on the right that doesn't factor, and I'm not sure what to do with it. I run into a similar problem if I try to isolate y' on the right first, and I'm not sure how to procede from this point.

The solution provided in the book is:

$y'=(y-4y^2)/(x+3+2y^3)$

Getting close, but not quite there.

 

[Dick]

Homework Statement

I am struggling with implicit derivatives, and though my course book includes final solutions to odd numbered exercises, it does not show the work. As such, I'm stuck in the process of getting from point A to point B:

Find the derivative y'(x) implicitly of

$((x+3)/y)=4x+y^2$

2. The attempt at a solution

I know to start with the derivative of each side of the equation:

$((d/dx)[x+3]y-(x+3)(d/dx)[y])/y^2=(d/dx)[4x+y^2]$
$((1)y-(x+3)(y'))/y^2=4+2y(y')$

Multiply both sides by $y^2$

$y-y'(x+3)=4y^2+2y^3(y')$

Subtract $y$ and then divide both sides by $-(x+3)$

$-y'(x+3)=4y^2-y+2y^3(y')$
$y'=-(4y^2-y+2y^3(y'))/(x+3)$

So now I have y' isolated on the left, but I still have a y' on the right that doesn't factor, and I'm not sure what to do with it. I run into a similar problem if I try to isolate y' on the right first, and I'm not sure how to procede from this point.

The solution provided in the book is:

$y'=(y-4y^2)/(x+3+2y^3)$

Getting close, but not quite there.

Starting from $y-y'(x+3)=4y^2+2y^3(y')$ move all of the terms containing y' to one side and factor y' out before you do anything else.