Derivative of 4√x Step by Step

Step by step

The question

Question: find the derivative of f(x) = 4 √x.

f(x) = 4√x

1 The rule

From the definition we found: x² gives 2x, and x³ gives 3x².

x² → 2x

x³ → 3x²

See the pattern?

The power comes down in front, and the power goes down by 1.

This is the power rule.

ddxxⁿ = nx^{n − 1}

We write the derivative as f′(x), "f prime of x".

f′(x): the derivative of f(x)

2 Write as powers

To use the rule, every x needs a power.

f(x) = 4x^(1/2)

A square root is the power one half.

√x = x^(1/2)

3 Each term

Use the rule on each term.

A number in front just waits, and multiplies.

f′(x) = 4 × 12x^(1/2 − 1)

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

12 − 1 = −12

Work out each one.

= 2x^(−1/2)

4 Tidy up

Books write it without minus or fraction powers.

So f′(x) = 2 over √x.

= 2√x

5 Check

Let's check: the derivative is the slope.

At x = 1, the rule gives 2.

f′(1) = 2

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

A calculator gives about 2.

Very close to 2.

Right!

f(1.001) − f(1)0.001 ≈ 2

Let's look back.

First: the rule.

Then: write as powers.

Then: each term.

Then: tidy up.

Last: check.

Answer