Step by step
The question
Question: find dy/dx for y² = 4x.
y2 = 4x
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.
y depends on x: y2 → 2y × dydx
1 The derivative of each side
Now each term.
x terms as usual, y terms times dy/dx, a number alone gives 0.
2y dydx = 4
Careful!
y depends on x.
So we must multiply by dy/dx.
Careful: not y2 → 2y ✗
2 Get dy/dx alone
To remove times 2y, we divide both sides by it.
dydx = 42y
Every number on the top and the bottom divides by 2.
top and bottom ÷ 2
So dy/dx = 2 over y.
dydx = 2y
3 Check
Let's check at a point on the curve: (1, 2).
Our dy/dx gives 1 there.
at (1, 2): dy/dx = 1
Now a very close point on the curve.
At x = 1.001, a calculator gives y ≈ 2.001.
x = 1.001: y ≈ 2.001
The slope between the two points: up 0.001, across 0.001.
About 1.
Very close to 1.
Right!
2.001 − 20.001 ≈ 1
Let's look back.
First: the derivative of each side.
Then: get dy/dx alone.
Last: check.
Answer
dydx = 2y