Differentiating Powers of x (Including Rational Powers) — Edexcel A-level Mathematics (Pure)

SPEC 7.2 • DIFFERENTIATION

DIFFERENTIATING POWERS OF x

The power rule for any real exponent

The Power Rule

If f (x) = x^{n}, then f^{'} (x) = n x^{n - 1}

Method

"Bring the power down, reduce the power by 1"

This rule works for any real value of n:

Positive integers:

x^{2}, x^{3}, x^{10}

Negative integers:

x^{- 1}, x^{- 2}, x^{- 3}

Fractions:

x^{1/2}, x^{3/2}, x^{- 1/2}

With Coefficients

If

f (x) = a x^{n}

, then

f^{'} (x) = an x^{n - 1}

Special Cases

\frac{d}{d x} (x) = 1

\frac{d}{d x} (c) = 0

01Worked Example

Differentiate:

(a) $x^{6}$ (b) $3 x^{4}$ (c) $5 x^{3} + 2 x^{2} - 7 x + 4$

(a)

f (x) = x^{6}

f^{'} (x) = 6 x^{6 - 1} = 6 x^{5}

(b)

f (x) = 3 x^{4}

f^{'} (x) = 3 \times 4 x^{4 - 1} = 12 x^{3}

(c)

y = 5 x^{3} + 2 x^{2} - 7 x + 4

Differentiate term by term:

\frac{d y}{d x} = 15 x^{2} + 4 x - 7

(Note: the constant 4 differentiates to 0)

Negative Powers

To differentiate expressions like

\frac{1}{x ^{n}}

, first convert to index form:

Key Conversions

\frac{1}{x} = x^{- 1}

\frac{1}{x ^{2}} = x^{- 2}

\frac{1}{x ^{3}} = x^{- 3}

Then apply the power rule as normal:

\frac{d}{d x} (x^{- 2}) = - 2 x^{- 2 - 1} = - 2 x^{- 3} = \frac{- 2}{x ^{3}}

Watch Your Arithmetic

When the power is negative, subtracting 1 makes it more negative:

- 2 - 1 = - 3

(not

- 1

!)

02Worked Example

Differentiate:

(a) $x^{- 3}$ (b) $\frac{4}{x ^{2}}$ (c) $x^{2} - \frac{3}{x} + \frac{2}{x ^{3}}$

(a)

f (x) = x^{- 3}

f^{'} (x) = - 3 x^{- 3 - 1} = - 3 x^{- 4} = \frac{- 3}{x ^{4}}

(b)

f (x) = \frac{4}{x ^{2}} = 4 x^{- 2}

f^{'} (x) = 4 \times (- 2) x^{- 3} = - 8 x^{- 3} = \frac{- 8}{x ^{3}}

(c)

y = x^{2} - \frac{3}{x} + \frac{2}{x ^{3}} = x^{2} - 3 x^{- 1} + 2 x^{- 3}

\frac{d y}{d x} = 2 x - 3 (- 1) x^{- 2} + 2 (- 3) x^{- 4}

= 2 x + 3 x^{- 2} - 6 x^{- 4} = 2 x + \frac{3}{x ^{2}} - \frac{6}{x ^{4}}

Fractional Powers

Roots can be written as fractional powers:

Key Conversions

x = x^{1/2}

3 x = x^{1/3}

4 x = x^{1/4}

x^{3} = x^{3/2}

3 x^{2} = x^{2/3}

\frac{1}{x} = x^{- 1/2}

Then apply the power rule:

\frac{d}{d x} (x) = \frac{d}{d x} (x^{1/2}) = \frac{1}{2} x^{- 1/2} = \frac{1}{2 x}

Fractional Arithmetic

\frac{1}{2} - 1 = \frac{1}{2} - \frac{2}{2} = - \frac{1}{2}

\frac{3}{2} - 1 = \frac{3}{2} - \frac{2}{2} = \frac{1}{2}

03Worked Example

Differentiate:

(a) $x$ (b) $x^{3/2}$ (c) $4 3 x$

(a)

f (x) = x = x^{1/2}

f^{'} (x) = \frac{1}{2} x^{1/2 - 1} = \frac{1}{2} x^{- 1/2} = \frac{1}{2 x}

(b)

f (x) = x^{3/2}

f^{'} (x) = \frac{3}{2} x^{3/2 - 1} = \frac{3}{2} x^{1/2} = \frac{3 x}{2}

(c)

f (x) = 4 3 x = 4 x^{1/3}

f^{'} (x) = 4 \times \frac{1}{3} x^{1/3 - 1} = \frac{4}{3} x^{- 2/3} = \frac{4}{3 3 x ^{2}}

04Worked Example

Differentiate:

(a) $\frac{6}{x}$ (b) $\frac{x ^{2} + 3}{x}$ (c) $x (x + 1)$

(a)

f (x) = \frac{6}{x} = 6 x^{- 1/2}

f^{'} (x) = 6 \times (- \frac{1}{2}) x^{- 3/2} = - 3 x^{- 3/2} = \frac{- 3}{x ^{3}}

(b) First simplify:

\frac{x ^{2} + 3}{x} = \frac{x ^{2}}{x} + \frac{3}{x} = x + 3 x^{- 1}

\frac{d y}{d x} = 1 + 3 (- 1) x^{- 2} = 1 - \frac{3}{x ^{2}}

(c) First expand:

x (x + 1) = x^{1/2} \cdot x + x^{1/2} = x^{3/2} + x^{1/2}

\frac{d y}{d x} = \frac{3}{2} x^{1/2} + \frac{1}{2} x^{- 1/2} = \frac{3 x}{2} + \frac{1}{2 x}

Practice

Question 1: Differentiate:

(a)

x^{8}

(b)

4 x^{5}

(c)

x^{3} + 3 x^{2} - 7 x + 2

(a)

\frac{d}{d x} (x^{8}) = 8 x^{7}

(b)

\frac{d}{d x} (4 x^{5}) = 20 x^{4}

(c)

\frac{d}{d x} (x^{3} + 3 x^{2} - 7 x + 2) = 3 x^{2} + 6 x - 7

Practice

Question 2: Differentiate:

(a)

\frac{1}{x ^{4}}

(b)

\frac{5}{x ^{2}}

(c)

x^{2} - \frac{3}{x}

(a)

\frac{1}{x ^{4}} = x^{- 4}

, so

\frac{d}{d x} = - 4 x^{- 5} = \frac{- 4}{x ^{5}}

(b)

\frac{5}{x ^{2}} = 5 x^{- 2}

, so

\frac{d}{d x} = - 10 x^{- 3} = \frac{- 10}{x ^{3}}

(c)

x^{2} - \frac{3}{x} = x^{2} - 3 x^{- 1}

\frac{d}{d x} = 2 x + 3 x^{- 2} = 2 x + \frac{3}{x ^{2}}

Practice

Question 3: Differentiate:

(a)

x

(b)

6 x

(c)

x x

(a)

x = x^{1/2}

, so

\frac{d}{d x} = \frac{1}{2} x^{- 1/2} = \frac{1}{2 x}

(b)

6 x = 6 x^{1/2}

, so

\frac{d}{d x} = 3 x^{- 1/2} = \frac{3}{x}

(c)

x x = x \cdot x^{1/2} = x^{3/2}

\frac{d}{d x} = \frac{3}{2} x^{1/2} = \frac{3 x}{2}

Summary

Key Rules

FunctionDerivative

x^{n}

n x^{n - 1}

a x^{n}

an x^{n - 1}

constant

c

0

Key Conversions

\frac{1}{x ^{n}} = x^{- n}

x = x^{1/2}

,

3 x = x^{1/3}

x^{m} = x^{m /2}

,

n x^{m} = x^{m / n}

Exam Tip

Always convert to index form before differentiating! Rewrite roots as fractional powers and fractions as negative powers first.