Derivative of √(2x + 1) Step by Step

Step by step

The question

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

y = √2x + 1

1 The inside

There is something inside: 2x + 1.

Call it u.

u = 2x + 1

Then y is just u^(1/2).

Written as a power of u.

y = u^(1/2)

A square root is the power one half.

√u = u^(1/2)

2 The outside

First the outside.

Use the power rule on u^(1/2).

dydu = 12u^(1/2 − 1)

We get 1/2 × u^(−1/2).

= 12u^(−1/2)

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

12 − 1 = −12

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.

= 12(2x + 1)^(−1/2) × 2

Careful!

Do not stop here!

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

Careful: not 12(2x + 1)^(−1/2) ✗

Multiply the numbers in front.

Then write the minus power at the bottom.

So dy/dx = 1 over √(2x + 1).

= 1√2x + 1

12 × 2 = 1

5 Check

Let's check: the derivative is the slope.

At x = 0, the rule gives 1.

dy/dx at x = 0: 1

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

A calculator gives about 1.

Very close to 1.

Right!

y(0.001) − y(0)0.001 ≈ 1

Let's look back.

First: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.

Answer