# VCE Mathematical Methods/Differentiation from First Principles

## Contents

## Theory[edit]

### Formula[edit]

Given a function *f*, the rule of the **derivative** (sometimes called the "gradient") function is defined as .

### Method[edit]

Remember that in order to evaluate a limit, we usually substitute the value given into the expression. However, with the above formula, substituting will result in a division by zero, which is mathematically impossible. Therefore,in order to make use of this formula, you need to substitute the rules and , then simplify to eliminate the fraction, and only then substitute . This is called **differentiation from first principles**.

For example:

Let

Let us differentiate *f* from first principles.

.

Therefore, we can define the gradient function as

## Exercises[edit]

**Question One**

Differentiate the following functions from first principles.

**(a)**

**(b)**

**(c)**

**(d)**

**Question Two**

Differentiate the following functions from first principles.

**(a)**

**(b)**

**(c)**

**(d)**