x^2 + xy + y^2 = 7, find the slope at (1, 2) Step by Step

Step by step

The question

Question: find the slope of x² + xy + y² = 7 at the point (1, 2).

x² + xy + y² = 7

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.

y depends on x: y² → 2y × dydx

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.

(uv)′ = u′v + uv′

xy → y + x dydx

Now each term.

x terms as usual, y terms times dy/dx, a number alone gives 0.

2x + y + x dydx + 2y dydx = 0

Careful!

y depends on x.

So we must multiply by dy/dx.

Careful: not y² → 2y ✗

2 Get dy/dx alone

Keep the dy/dx terms on the left.

Move the others to the right: they change sign.

x dydx + 2y dydx = −2x − y

Two terms have dy/dx.

Take dy/dx out of both.

dydx(x + 2y) = −2x − y

To remove times (x + 2y), we divide both sides by it.

dydx = −2x − yx + 2y

Tidy up: dy/dx = −(2x + y)/(x + 2y).

dydx = −2x + yx + 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.

1² + (1)(2) + 2² = 1 + 2 + 4 = 7 ✓

Put x = 1 and y = 2 into dy/dx.

dydx = −2(1) + 21 + 2(2)

So the slope at (1, 2) is −4/5.

= −45

4 Check

Let's check with a very close point on the curve.

At x = 1.001, a calculator gives y ≈ 1.9992.

x = 1.001: y ≈ 1.9992

The slope between the two points: down 0.0008, across 0.001.

About −0.8.

Very close to −0.8.

Right!

1.9992 − 20.001 ≈ −0.8

Let's look back.

First: the derivative of each side.

Then: get dy/dx alone.

Then: the slope at the point.

Last: check.

Answer