Step by step
The question
Question: find the slope of x² + y² = 25 at the point (3, 4).
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
Keep the dy/dx terms on the left.
Move the others to the right: they change sign.
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 = −x over y.
3 The slope at the point
First, is (3, 4) on the curve?
Put x = 3 and y = 4 in.
We get 25.
Yes, it is.
Put x = 3 and y = 4 into dy/dx.
So the slope at (3, 4) is −3/4.
4 Check
Let's check with a very close point on the curve.
At x = 3.001, a calculator gives y ≈ 3.99925.
The slope between the two points: down 0.00075, across 0.001.
About −0.75.
Very close to −0.75.
Right!
Let's look back.
First: the derivative of each side.
Then: get dy/dx alone.
Then: the slope at the point.
Last: check.