Derivative of 3√(2x + 5) Step by Step

Step by step

The question

Question: find dy/dx for y = 3 √(2x + 5).

y = 3√2x + 5

1 The inside

There is something inside: 2x + 5.

Call it u.

u = 2x + 5

Then y is just 3 u^(1/2).

Written as a power of u.

y = 3u^(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 3 u^(1/2).

dydu = 3 × 12u^(1/2 − 1)

We get 3/2 × u^(−1/2).

= 32u^(−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 + 5 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.

= 32(2x + 5)^(−1/2) × 2

Careful!

Do not stop here!

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

Careful: not 32(2x + 5)^(−1/2) ✗

Multiply the numbers in front.

Then write the minus power at the bottom.

So dy/dx = 3 over √(2x + 5).

= 3√2x + 5

32 × 2 = 3

5 Check

Let's check: the derivative is the slope.

At x = 0, the rule gives 1.3416.

dy/dx at x = 0: 1.3416

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

A calculator gives about 1.34.

Very close to 1.3416.

Right!

y(0.001) − y(0)0.001 ≈ 1.34

Let's look back.

First: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.

Answer