Step by step
The question
Question: find the slope of x² + y² = 169 at the point (5, 12).
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 (5, 12) on the curve?
Put x = 5 and y = 12 in.
We get 169.
Yes, it is.
Put x = 5 and y = 12 into dy/dx.
So the slope at (5, 12) is −5/12.
4 Check
Let's check with a very close point on the curve.
At x = 5.001, a calculator gives y ≈ 11.99958.
The slope between the two points: down 0.00042, across 0.001.
About −0.42.
Very close to −0.4167.
Right!
Let's look back.
First: the derivative of each side.
Then: get dy/dx alone.
Then: the slope at the point.
Last: check.