If f(x) = (x^2 - 5)^3, what is f′(x)? Step by Step

Step by step

The question

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

y = (x² − 5)³

There are four choices.

First we solve it, the normal way.

Then we pick.

1 The inside

There is something inside: x² − 5.

Call it u.

u = x² − 5

Then y is just u³.

y = u³

2 The outside

First the outside.

Use the power rule on u³.

dydu = 3u^(3 − 1)

We get 3 u².

= 3u²

3 The inside's slope

Now the inside.

The power rule on x² − 5 gives 2x.

dudx = 2x

4 Multiply

The chain rule: the outside's derivative times the inside's derivative, dy/du times du/dx.

dydx = dydu × dudx

Put the inside back in place of u.

= 3(x² − 5)² × 2x

Careful!

Do not stop here!

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

Careful: not 3(x² − 5)² ✗

Multiply the numbers in front: 3 times 2x is 6x.

3 × 2x = 6x

So dy/dx = 6x (x² − 5)².

= 6x(x² − 5)²

5 Check

Let's check: the derivative is the slope.

At x = 1, the rule gives 96.

dy/dx at x = 1: 96

Now the slope between two very close points: x = 1 and x = 1.001.

A calculator gives about 96.

Very close to 96.

Right!

y(1.001) − y(1)0.001 ≈ 96

6 Pick the choice

Our answer is dy over dx = 6x(x² − 5)².

That is choice C.

C) 6x(x² − 5)²

7 Why not the others

A is 3(x² − 5)².

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

That is the Careful mistake from before.

A) 3(x² − 5)² ✗ lost × 2x, the derivative of the inside

B is 6x(x² − 5)³.

Wrong: the power goes down by 1.

B) 6x(x² − 5)³ ✗ the power goes down by 1

D is 2x(x² − 5)².

Wrong: the power 3 comes down in front too.

D) 2x(x² − 5)² ✗ the power 3 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.

Answer