y^3 = x^2 + 7, find dy/dx

Step by step

The question

Question: find dy/dx for y³ = x² + 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.

1 The derivative of each side

So y³ gives 3y² 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

To remove times 3y², we divide both sides by it.

So dy/dx = 2x/3y².

3 Check

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

Our dy/dx gives 0.1667 there.

Now a very close point on the curve.

At x = 1.001, a calculator gives y ≈ 2.00017.

The slope between the two points: up 0.00017, across 0.001.

About 0.17.

Very close to 0.1667.

Right!

Let's look back.

First: the derivative of each side.

Then: get dy/dx alone.

Last: check.