Derivative of 1/(x + 5)^3 Step by Step

Step by step

The question

Question: find dy/dx for y = 1/(x + 5)³.

y = 1(x + 5)³

1 The inside

There is something inside: x + 5.

Call it u.

u = x + 5

Then y is just u⁻³.

Written as a power of u.

y = u⁻³

u to a power at the bottom is a minus power on top.

1u³ = u⁻³

2 The outside

First the outside.

Use the power rule on u⁻³.

dydu = (−3)u^(−3 − 1)

We get −3 u⁻⁴.

= −3u⁻⁴

Take 1 away from the power: −3 − 1 = −4.

−3 − 1 = −4

3 The inside's slope

Now the inside.

The power rule on x + 5 gives 1.

dudx = 1

4 Multiply

The chain rule: multiply the two.

dy/du times du/dx.

dydx = dydu × dudx

Put the inside back in place of u.

= −3(x + 5)⁻⁴ × 1

Careful!

Do not stop here!

We must multiply by the inside's slope, 1.

Careful: not −3(x + 5)⁻⁴ ✗

Multiply the numbers in front.

Then write the minus power at the bottom.

So dy/dx = −3/(x + 5)⁴.

= −3(x + 5)⁴

−3 × 1 = −3

5 Check

Let's check: the derivative is the slope.

At x = 0, the rule gives −0.0048.

dy/dx at x = 0: −0.0048

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

A calculator gives about 0.

Very close to −0.0048.

Right!

y(0.001) − y(0)0.001 ≈ 0

Let's look back.

First: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.

Answer