Derivative of 6x^-3 Step by Step

Step by step

The question

Question: find the derivative of f(x) = 6 x⁻³.

f(x) = 6x⁻³

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 Each term

Use the rule on each term.

A number in front just waits, and multiplies.

f′(x) = 6 × (−3)x^(−3 − 1)

Take 1 away from the power: −3 − 1 = −4.

−3 − 1 = −4

Careful!

Taking 1 away from −3 makes it −4.

The power goes down, never up.

Careful: not x⁻³ → x⁻² ✗

Work out each one.

= −18x⁻⁴

3 Tidy up

Books write it without minus or fraction powers.

So f′(x) = −18/x⁴.

= −18x⁴

4 Check

Let's check: the derivative is the slope.

At x = 1, the rule gives −18.

f′(1) = −18

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

A calculator gives about −17.96.

Very close to −18.

Right!

f(1.001) − f(1)0.001 ≈ −17.96

Let's look back.

First: the rule.

Then: each term.

Then: tidy up.

Last: check.

Answer