Derivative of 8/x^4 - x^2 Step by Step

Step by step

The question

Question: find the derivative of f(x) = 8/x⁴ − x².

f(x) = 8x⁴ − x²

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) = 8x⁻⁴ − x²

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

1x⁴ = x⁻⁴

3 Each term

Use the rule on each term.

A number in front just waits, and multiplies.

f′(x) = 8 × (−4)x^(−4 − 1) − 2x^(2 − 1)

Take 1 away from the power: −4 − 1 = −5.

−4 − 1 = −5

Careful!

Taking 1 away from −4 makes it −5.

The power goes down, never up.

Careful: not x⁻⁴ → x⁻³ ✗

Work out each one.

= −32x⁻⁵ − 2x

4 Tidy up

Books write it without minus or fraction powers.

So f′(x) = −32/x⁵ − 2x.

= −32x⁵ − 2x

5 Check

Let's check: the derivative is the slope.

At x = 1, the rule gives −34.

f′(1) = −32 − 2 = −34

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

A calculator gives about −33.92.

Very close to −34.

Right!

f(1.001) − f(1)0.001 ≈ −33.92

Let's look back.

First: the rule.

Then: write as powers.

Then: each term.

Then: tidy up.

Last: check.

Answer