Derivative of 4/x^2 Step by Step

Step by step

The question

Question: find the derivative of f(x) = 4/x².

f(x) = 4x²

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⁻²

x to a power at the bottom is the same as a minus power on top.

1x² = x⁻²

3 Each term

Use the rule on each term.

A number in front just waits, and multiplies.

f′(x) = 4 × (−2)x^(−2 − 1)

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

−2 − 1 = −3

Careful!

Taking 1 away from −2 makes it −3.

The power goes down, never up.

Careful: not x⁻² → x⁻¹ ✗

Work out each one.

= −8x⁻³

4 Tidy up

Books write it without minus or fraction powers.

So f′(x) = −8/x³.

= −8x³

5 Check

Let's check: the derivative is the slope.

At x = 1, the rule gives −8.

f′(1) = −8

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

A calculator gives about −7.99.

Very close to −8.

Right!

f(1.001) − f(1)0.001 ≈ −7.99

Let's look back.

First: the rule.

Then: write as powers.

Then: each term.

Then: tidy up.

Last: check.

Answer