# Differentiation

## Differentiation Revision

**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}

**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

**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

**Differentiating Negative Roots**

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

**Example:**

**Differentiate** 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}

**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}

**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) = 2x + 3

## Differentiation Example Questions

**Question 1: **Write the derivative of y=2x^4 with respect to x.

**[2 marks]**

\dfrac{\text{d}y}{\text{d}x} = 8x^3

**Question 2: **Write the derivative of f(x)=x^{-3} with respect to x.

**[2 marks]**

**Question 3: **Write the derivative of f(x)=(x+4)(x+6) - x^3 with respect to x.

**[3 marks]**

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