Step by step
The question
Question: find dy/dx for xy + 2x = 8.
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.
2 Get dy/dx alone
Keep the dy/dx terms on the left.
Move the others to the right: they change sign.
To remove times x, we divide both sides by it.
So dy/dx is minus the fraction (y + 2) over x.
3 Check
Let's check at a point on the curve: (2, 2).
Our dy/dx gives −2 there.
Now a very close point on the curve.
At x = 2.001, a calculator gives y ≈ 1.998.
The slope between the two points: down 0.002, across 0.001.
About −2.
Very close to −2.
Right!
Let's look back.
First: the derivative of each side.
Then: get dy/dx alone.
Last: check.