Step by step
The question
Question: find the slope of x² − y² = 9 at the point (5, 4).
The derivative when y is mixed in with x.
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.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 The derivative of each side
So −y² gives −2y 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
We want the dy/dx terms alone on the left.
So we take away 2x from both sides.
To remove times −2y, we divide both sides by it.
The numbers are all 2, so we divide by 2.
Tidy up: dy/dx = x over y.
3 The slope at the point
First, is (5, 4) on the curve?
Put x = 5 and y = 4 in.
We get 9.
Yes, it is.
Put x = 5 and y = 4 into dy/dx.
So the slope at (5, 4) is 5/4.
4 Check
Let's check with a very close point on the curve.
At x = 5.001, a calculator gives y ≈ 4.00125.
The slope between the two points: up 0.00125, across 0.001.
About 1.25.
Very close to 1.25.
Right!
5 Pick the choice
Our answer is slope = 5/4.
That is choice C.
6 Why not the others
A is 4/5.
Wrong: y/x: it is x over y.
B is −5/4.
Wrong: the minus: y² has a minus, so the slope is +x/y.
D is 1.
Wrong: used the change 5 − 4.
Let's look back.
First: the derivative of each side.
Then: get dy/dx alone.
Then: the slope at the point.
Then: check.
Then: pick the choice.
Last: why not the others.