Introduction to Derivatives (Edexcel IGCSE Further Maths)

What is differentiation?

• Differentiation is part of the branch of mathematics called calculus
• It is concerned with the rate at which changes takes place
  • So it has many real‑world applications:
    • The rate at which a car is moving (its speed)
    • The rate at which a virus spreads amongst a population

 

• To begin to understand differentiation you’ll need to understand gradients

How are gradients related to rates of change?

• Gradient in everyday language refers to steepness.
  • For example, the gradient of a road up the side of a hill is important to lorry drivers
• On a graph the gradient refers to how 'steep' a line or a curve is
• • It is a way of measuring how fast y  changes as x  changes
    • This may be referred to as the 'rate of change of y  with respect to x'
• So gradient describes the rate at which change happens

How do I find the gradient of a curve using its graph?

For a straight line the gradient is always the same (constant)

• Recall  y = mx + c, where m  is the gradient

• For a curve the gradient changes as the value of x  changes
  • A tangent is a straight line that touches the curve at one point
    • At any point on the curve, the gradient of the curve is equal to the gradient of the tangent at that point

How do I find the gradient of a curve using calculus? 

• The equation of a curve can be given in the form
  • Substituting x-coordinate inputs gives y-coordinate outputs
• It is possible to create an algebraic function that take inputs of x-coordinates
  • and gives outputs that are gradients
    • No graph sketching required!
• This type of function has a few commonly used names:
  
  • The gradient function
  • The derivative
  • The derived function
• The way to write this function is 
  • This is pronounced "dy by dx" or "dy over dx"
  • In function notation, it can be written 
    • pronounced "f-dash-of-x" or "f-prime-of-x"
• To get from  to  you need to do an operation called differentiation
  • Differentiation turns curve equations into gradient functions
  • There are standard formulae used to differentiate all the basic functions
    • See the 'Differentiating Basic Functions' revision note
  • There are also various methods for differentiating more complicated functions
    • See the 'Techniques of Differentiation' revision note
• Once you know  for a curve, you can find the gradient for any point on the curve
  • gives the value of the gradient when 
    • See the 'Finding Gradients' revision note for more details