# VCE Mathematical Methods/Differentiation from First Principles

## TheoryEdit

### FormulaEdit

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

### MethodEdit

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

## ExercisesEdit

**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)**