If f(x) = (2x + 1)^5, what is f′(x)?

Step by step

The question

Question: find dy/dx for y = (2x + 1)⁵.

There are four choices.

First we solve it, the normal way.

Then we pick.

1 The inside

There is something inside: 2x + 1.

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 2x + 1 gives 2.

4 Multiply

The chain rule: the outside's derivative times the inside's derivative, 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, 2.

Multiply the numbers in front: 5 times 2 is 10.

So dy/dx = 10 (2x + 1)⁴.

5 Check

Let's check: the derivative is the slope.

At x = 0, the rule gives 10.

Now the slope between two very close points: x = 0 and x = 0.001.

A calculator gives about 10.04.

Very close to 10.

Right!

6 Pick the choice

Our answer is dy over dx = 10(2x + 1)⁴.

That is choice B.

7 Why not the others

A is 5(2x + 1)⁴.

Wrong: lost × 2, the derivative of the inside.

That is the Careful mistake from before.

C is 10(2x + 1)⁵.

Wrong: the power goes down by 1.

D is 2(2x + 1)⁴.

Wrong: the power 5 comes down in front too.

Let's look back.

First: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Then: check.

Then: pick the choice.

Last: why not the others.