Step by step
The question
Question: find the slope of x² + xy + y² = 7 at the point (1, 2).
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
xy is x times y: use the product rule.
The derivative of x is 1, and of y is 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 and take away y from both sides.
Two terms have dy/dx.
Take dy/dx out of both.
To remove times (x + 2y), we divide both sides by it.
Tidy up: dy/dx = −(2x + y)/(x + 2y).
3 The slope at the point
First, is (1, 2) on the curve?
Put x = 1 and y = 2 in.
We get 7.
Yes, it is.
Put x = 1 and y = 2 into dy/dx.
So the slope at (1, 2) is −4/5.
4 Check
Let's check with a very close point on the curve.
At x = 1.001, a calculator gives y ≈ 1.9992.
The slope between the two points: down 0.0008, across 0.001.
About −0.8.
Very close to −0.8.
Right!
5 Pick the choice
Our answer is slope = −4/5.
That is choice B.
6 Why not the others
A is 4/5.
Wrong: lost the minus sign.
C is −5/4.
Wrong: turned the fraction upside down.
D is −1/2.
Wrong: lost the xy term: its derivative is y + x dy/dx.
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.