Derivative of 4/√(x + 1) Step by Step

Step by step

The question

Question: find dy/dx for y = 4 √(x + 1).

y = 4√x + 1

1 The inside

There is something inside: x + 1.

Call it u.

u = x + 1

Then y is just 4 u^(−1/2).

Written as a power of u.

y = 4u^(−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 4 u^(−1/2).

dydu = 4 × 12u^(−1/2 − 1)

We get −2 u^(−3/2).

= −2u^(−3/2)

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

−12 − 1 = −32

3 The inside's slope

Now the inside.

The power rule on x + 1 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.

= −2(x + 1)^(−3/2) × 1

Careful!

Do not stop here!

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

Careful: not −2(x + 1)^(−3/2) ✗

Multiply the numbers in front.

Then write the minus power at the bottom.

So dy/dx = −2/(x + 1)^(3/2).

= −2(x + 1)^(3/2)

−2 × 1 = −2

5 Check

Let's check: the derivative is the slope.

At x = 0, the rule gives −2.

dy/dx at x = 0: −2

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

A calculator gives about −2.

Very close to −2.

Right!

y(0.001) − y(0)0.001 ≈ −2

Let's look back.

First: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.

Answer