Derivative of 3/x Step by Step

Step by step

The question

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

f(x) = 3x

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) = 3x⁻¹

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

1x¹ = x⁻¹

3 Each term

Use the rule on each term.

A number in front just waits, and multiplies.

f′(x) = 3 × (−1)x^(−1 − 1)

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

−1 − 1 = −2

Careful!

Taking 1 away from −1 makes it −2.

The power goes down, never up.

Careful: not x⁻¹ → x⁰ ✗

Work out each one.

= −3x⁻²

4 Tidy up

Books write it without minus or fraction powers.

So f′(x) = −3/x².

= −3x²

5 Check

Let's check: the derivative is the slope.

At x = 1, the rule gives −3.

f′(1) = −3

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

A calculator gives about −3.

Very close to −3.

Right!

f(1.001) − f(1)0.001 ≈ −3

Let's look back.

First: the rule.

Then: write as powers.

Then: each term.

Then: tidy up.

Last: check.

Answer