Derivative of (2x + 3)^(-2) Step by Step

Step by step

The question

Question: find dy/dx for y = (2x + 3)⁻².

y = (2x + 3)⁻²

1 The inside

There is something inside: 2x + 3.

Call it u.

u = 2x + 3

Then y is just u⁻².

y = u⁻²

2 The outside

First the outside.

Use the power rule on u⁻².

dydu = (−2)u^(−2 − 1)

We get −2 u⁻³.

= −2u⁻³

Take 1 away from the power: −2 − 1 = −3.

−2 − 1 = −3

3 The inside's slope

Now the inside.

The power rule on 2x + 3 gives 2.

dudx = 2

4 Multiply

The chain rule: multiply the two.

dy/du times du/dx.

dydx = dydu × dudx

Put the inside back in place of u.

= −2(2x + 3)⁻³ × 2

Careful!

Do not stop here!

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

Careful: not −2(2x + 3)⁻³ ✗

Multiply the numbers in front.

Then write the minus power at the bottom.

So dy/dx = −4/(2x + 3)³.

= −4(2x + 3)³

−2 × 2 = −4

5 Check

Let's check: the derivative is the slope.

At x = 0, the rule gives −0.1481.

dy/dx at x = 0: −0.1481

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

A calculator gives about −0.15.

Very close to −0.1481.

Right!

y(0.001) − y(0)0.001 ≈ −0.15

Let's look back.

First: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.

Answer