y^2 + 3y = x^2, find dy/dx

Step by step

The question

Question: find dy/dx for y² + 3y = x².

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

Two terms have dy/dx.

Take dy/dx out of both.

To remove times (2y + 3), we divide both sides by it.

So dy/dx = 2x/(2y + 3).

3 Check

Let's check at a point on the curve: (2, 1).

Our dy/dx gives 0.8 there.

Now a very close point on the curve.

At x = 2.001, a calculator gives y ≈ 1.0008.

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

About 0.8.

Very close to 0.8.

Right!

Let's look back.

First: the derivative of each side.

Then: get dy/dx alone.

Last: check.