Step by step
The question
Question: find the slope of x² + y² = 100 at the point (6, −8).
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
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 (6, −8) on the curve?
Put x = 6 and y = −8 in.
We get 100.
Yes, it is.
Put x = 6 and y = −8 into dy/dx.
So the slope at (6, −8) is 3/4.
4 Check
Let's check with a very close point on the curve.
At x = 6.001, a calculator gives y ≈ −7.99925.
The slope between the two points: up 0.00075, across 0.001.
About 0.75.
Very close to 0.75.
Right!
5 Pick the choice
Our answer is slope = 3/4.
That is choice A.
6 Why not the others
B is 4/3.
Wrong: y/x: it is x over y.
C is −3/4.
Wrong: −x/y = −6/(−8): two minus signs make plus.
D is −4/3.
Wrong: y over x.
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.