Derivative of (x^2 - 3x)^5

Step by step

The question

Question: find dy/dx for y = (x² − 3x)⁵.

1 The inside

There is something inside: x² − 3x.

Call it u.

Then y is just u⁵.

2 The outside

First the outside.

Use the power rule on u⁵.

We get 5 u⁴.

3 The inside's slope

Now the inside.

The power rule on x² − 3x gives 2x − 3.

4 Multiply

The chain rule: multiply the two.

dy/du times du/dx.

Put the inside back in place of u.

Careful!

Do not stop here!

We must multiply by the inside's slope, 2x − 3.

So dy/dx = 5(2x − 3) (x² − 3x)⁴.

5 Check

Let's check: the derivative is the slope.

At x = undefined, the rule gives NaN.

Now the slope between two very close points: x = undefined and x = NaN.

A calculator gives about NaN.

Very close to NaN.

Right!

Let's look back.

First: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.