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Differentiation Revision


Differentiation is a method used to find the variable rate of change or the gradient of a line. When you differentiate, the result is the derivative, representing the rate of change or gradient.

Differentiation Notation & Basics

Notation 1:

Let’s differentiate the function y = x^{\textcolor{blue}{n}}

The derivative is \dfrac{\text{d}y}{\text{d}x}=\textcolor{blue}{n}x^{\textcolor{blue}{n}-1}


Differentiate y = 2x^\textcolor{blue}{3}

\dfrac{\text{d}y}{\text{d}x}=\textcolor{blue}{3}\times 2x^{\textcolor{blue}{3}-1} = 6x^2


Notation 2:

Let’s differentiate the function f(x) = x^{\textcolor{blue}{n}}

Our derivative, f'(x),=\textcolor{blue}{n}x^{\textcolor{blue}{n}-1}


Differentiate f(x) = 4x^\textcolor{blue}{9}

f'(x)=\textcolor{blue}{9}\times 4x^{\textcolor{blue}{9}-1} = 36x^8

Level 6-7GCSECambridge iGCSEEdexcel iGCSE

Differentiation of Linear Combinations

We can also use differentiation to find the derivative of functions involving multiple terms.

To do this, we simply differentiate each term individually:


Differentiate f(x) = 3x^2 + 4x + 17\\

As the 17 has no x value, we cannot differentiate it so we ignore it…

f'(x) = (\textcolor{blue}{2}\times 3x^{\textcolor{blue}{2}-1}) + (\textcolor{blue}{1}\times 4x^{\textcolor{blue}{1}-1}) + 0\\

f'(x) = 6x + 4

Level 6-7GCSECambridge iGCSEEdexcel iGCSE

Differentiating Negative Roots

We can differentiate functions with negative roots using the same method.


Differentiate y = 4x^{-2}

y = 4x^{-2}

Using \dfrac{\text{d}y}{\text{d}x}=\textcolor{blue}{n}x^{\textcolor{blue}{n}-1},

\dfrac{\text{d}y}{\text{d}x}=\textcolor{blue}{-2}\times 4x^{\textcolor{blue}{-2}-1}=-8x^{-3}

Level 6-7GCSEEdexcel iGCSE

Example 1: Differentiation

Differentiate the following expressions:

a) y=x^5

b) y=3x^2

c) y=x^{-7}

[6 marks]

a) \dfrac{\text{d}y}{\text{d}x}=\textcolor{blue}{5} \times x^{\textcolor{blue}{5}-1} = 5x^4

b) \dfrac{\text{d}y}{\text{d}x}=\textcolor{blue}{2} \times 3x^{\textcolor{blue}{2}-1} = 6x

c) \dfrac{\text{d}y}{\text{d}x}=\textcolor{blue}{-7} \times x^{\textcolor{blue}{-7}-1} = -7x^{-8}

Level 6-7GCSECambridge iGCSEEdexcel iGCSE

Example 2: Linear Combinations

Differentiate the function f(x) = (x+1)(x+2)

[3 marks]

To solve this, we will firstly expand the brackets:

f(x) = x^2 + 2x + x + 2\\ f(x) = x^2 + 3x + 2\\


We can now differentiate:

f'(x) = (\textcolor{blue}{2}\times x^{\textcolor{blue}{2}-1}) + (\textcolor{blue}{1}\times 3x^{\textcolor{blue}{1}-1}) + 0\\

f'(x) = 2x + 3

Level 6-7GCSECambridge iGCSEEdexcel iGCSE

Differentiation Example Questions



\dfrac{\text{d}y}{\text{d}x} = 8x^3
f'(x)=(1\times-3)x^{-3-1}\\ f'(x) = -3x^{-4}

Expand brackets:

f(x) = x^2 + 4x + 6x + 24 - x^3\\ f(x) = -x^3 + x^2 + 10x + 24 \\


Differentiate each term separately:

f'(x) = -3x^{2} + 2x + 10

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