Derivative of 3/(2x - 1) Step by Step

Step by step

The question

Question: find dy/dx for y = 3/(2x − 1).

y = 32x − 1

1 The inside

There is something inside: 2x − 1.

Call it u.

u = 2x − 1

Then y is just 3 u⁻¹.

Written as a power of u.

y = 3u⁻¹

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

1u¹ = u⁻¹

2 The outside

First the outside.

Use the power rule on 3 u⁻¹.

dydu = 3 × (−1)u^(−1 − 1)

We get −3 u⁻².

= −3u⁻²

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

−1 − 1 = −2

3 The inside's slope

Now the inside.

The power rule on 2x − 1 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.

= −3(2x − 1)⁻² × 2

Careful!

Do not stop here!

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

Careful: not −3(2x − 1)⁻² ✗

Multiply the numbers in front.

Then write the minus power at the bottom.

So dy/dx = −6/(2x − 1)².

= −6(2x − 1)²

−3 × 2 = −6

5 Check

Let's check: the derivative is the slope.

At x = 1, the rule gives −6.

dy/dx at x = 1: −6

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

A calculator gives about −5.99.

Very close to −6.

Right!

y(1.001) − y(1)0.001 ≈ −5.99

Let's look back.

First: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.

Answer