Step by step
The question
Question: find the slope of y² = 4x at the point (1, 2).
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
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
To remove times 2y, we divide both sides by it.
Every number on the top and the bottom divides by 2.
Tidy up: dy/dx = 2 over y.
3 The slope at the point
First, is (1, 2) on the curve?
Put x = 1 and y = 2 in.
Both sides give 4.
Yes, it is.
Put x = 1 and y = 2 into dy/dx.
So the slope at (1, 2) is 1.
4 Check
Let's check with a very close point on the curve.
At x = 1.001, a calculator gives y ≈ 2.001.
The slope between the two points: up 0.001, across 0.001.
About 1.
Very close to 1.
Right!
Let's look back.
First: the derivative of each side.
Then: get dy/dx alone.
Then: the slope at the point.
Last: check.