Step by step
The question
Question: find dy/dx for xy + y² = 12.
y is not written alone, but y still depends on x.
So for y we use the chain rule.
y squared gives 2y, times dy/dx.
1 The derivative of each side
xy is x times y: use the product rule.
The derivative of x is 1, and of y is dy/dx.
Now each term.
x terms as usual, y terms times dy/dx, a number alone gives 0.
Careful!
y depends on x.
So we must multiply by dy/dx.
2 Get dy/dx alone
Keep the dy/dx terms on the left.
Move the others to the right: they change sign.
Two terms have dy/dx.
Take dy/dx out of both.
To remove times (x + 2y), we divide both sides by it.
So dy/dx = −y/(x + 2y).
3 Check
Let's check at a point on the curve: (1, 3).
Our dy/dx gives −0.4286 there.
Now a very close point on the curve.
At x = 1.001, a calculator gives y ≈ 2.99957.
The slope between the two points: down 0.00043, across 0.001.
About −0.43.
Very close to −0.4286.
Right!
Let's look back.
First: the derivative of each side.
Then: get dy/dx alone.
Last: check.