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: