Step by step
The question
Question: find dy/dx for x³ + y³ = 9.
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
So y³ gives 3y² 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.
To remove times 3y², we divide both sides by it.
Every number on the top and the bottom divides by 3.
So dy/dx = −x² over y².
3 Check
Let's check at a point on the curve: (1, 2).
Our dy/dx gives −0.25 there.
Now a very close point on the curve.
At x = 1.001, a calculator gives y ≈ 1.99975.
The slope between the two points: down 0.00025, across 0.001.
About −0.25.
Very close to −0.25.
Right!
Let's look back.
First: the derivative of each side.
Then: get dy/dx alone.
Last: check.