Differentiation (CIE IGCSE Maths: Extended)

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 has lots of real‑world uses:
  • 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 generally means 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 really a way of measuring how fast y changes as x changes
• This may be referred to as the rate at which y

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

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

• This is really where the fun begins!
  • Drawing tangents each time you want the gradient of a curve is too much effort
  • It would be great if you could do it using algebra instead
• The equation of a curve can be given in the form
  • Inputting x-coordinates gives outputs of y-coordinates
• It is possible to create an algebraic function that take inputs of x-coordinates and gives outputs of gradients
  • All of this is done without needing to sketch any graphs
• 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"
  • In function notation, it can be written 
    • pronounced f-dashed-of-x
• To get from  to  you need to do an operation called differentiation
  • Differentiation turns curve equations into gradient functions
• The main rule for differentiation is shown

 

• This looks worse than it is!
• For powers of x

STEP 1   Multiply the number in front by the power

STEP 2   Take one off the power (reduce the power by 1)

• 2x⁶ differentiates to 12x⁵
  • Note the following:
    • kx differentiates to k
    • so 10x differentiates to 10
    • any number on its own differentiates to zero
    • so 8 differentiates to 0

 

How do I use the gradient function to find gradients of curves?

• Find the x-coordinate of the point on the curve you're interested in
• Use differentiation to find the gradient (derived) function,
• Substitute the x-coordinate into the gradient (derived) function to find the gradient