# Derivatives

## Prerequisites

### Derivatives

#### Introduction

• A derivitive is a function that describes another function.
• More specifically:
• It describes the gradient of a function.
• A gradient is how much y changes in response to a change in x.
• Finding the derivitive is sometimes called differentiation.
• Notation:
• If a function is given in the form:
• Then it's derivative is:
• This is pronounced "f prime of x".
• It is called "Lagrange's notation".
• if a function is given in the form:
• Then it's derivative is:
• This is pronounced "dy over dx" or "dy by dx".
• It is called "Leibniz's notation".
• It is also sometimes written as:

#### Derivative of a straight line

• We know that the equation of a straight line is:
• Where:
• m = the gradient of the slope.
• b = the point that the line intercepts the y axis.
• Straight lines are straight because their gradient doesn't change:
• For all points along the line, the gradient is the same. It is always m
• The derivative of a straight line is therefore:

#### Derivative of a curved line

• Curved lines are different to straight lines.
• Their gradient changes depending on where you measure it.
• for example: f(x) = x2
• If we examine the gradients at various points on the line:
• At x = -2, the gradient is -4
• At x = 1, the gradient is 2
• To find the gradient of at a certain point we:
• Choose a point that is very close to the original.
• Draw a straight line between them.
• And then find the gradient of the line.
• For our first point we can choose (x, y), or more specifically (x, x2).
• For our second point:
• We need a point very close to x
• Such as x + 0.1
• Or x + 0.01
• We use the greek letter delta to represent this value:
• Our second point is then:
• Our equation for the gradient is:
• Which simplifies to: And then:
• As delta approaches 0, the equation becomes: