# Differentiation

GCSELevel 6-7Cambridge iGCSEEdexcel iGCSE

## Differentiation

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

Example:

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

Example:

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:

Example:

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.

Example:

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

$\dfrac{\text{d}y}{\text{d}x}=(2\times4)x^{4-1}\\$

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

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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