5.1.1 Introduction to Differentiation

• Before introducing a derivative, an understanding of a limit is helpful

What is a limit?

• The limit of a function is the value a function (of ) approaches as  approaches a particular value from either side
  • Limits are of interest when the function is undefined at a particular value
  • For example, the function will approach a limit as approaches 1 from both below and above but is undefined at  as this would involve dividing by zero

What might I be asked about limits?

• You may be asked to predict or estimate limits from a table of function values or from the graph of 
• You may be asked to use your GDC to plot the graph and use values from it to estimate a limit

What is a derivative?

• Calculus is about rates of change
  • the way a car’s position on a road changes is its speed (velocity)
  • the way the car’s speed changes is its acceleration
• The gradient (rate of change) of a (non-linear) function varies with 
• The derivative of a function is a function that relates the gradient to the value of 
• The derivative is also called the gradient function

How are limits and derivatives linked?

• Consider the point  on the graph of  as shown below
  •  is a series of chords
• The gradient of the function at the point  is equal to the gradient of the tangent at point 
• The gradient of the tangent at point  is the limit of the gradient of the chords  as point  ‘slides’ down the curve and gets ever closer to point 
• The gradient of the function changes as  changes
• The derivative is the function that calculates the gradient from the value 

What is the notation for derivatives?

• For the function , the derivative, with respect to , would be written as

• Different variables may be used
  • e.g. If  then  

What might I be asked about derivatives?

• You may be asked to use the graphing features of your GDC to find the gradients of a function at different values of 
• From a series of gradient values, you may be asked to suggest an expression for the derivative (gradient function) of a function

What is differentiation?

• Differentiation is the process of finding an expression of the derivative (gradient function) from the expression of a function

How do I differentiate powers of x?

• Powers of  are differentiated according to the following formula:
  • If then where 
  • This is given in the formula booklet
• If the power of  is multiplied by a constant then the derivative is also multiplied by that constant
  • If then where  and  is a constant
• The alternative notation (to) is to use 
  • If  then 
    • e.g.  If  then 
• Don't forget these two special cases:
  • If then
    • e.g.  If  then 
  • If then
    • e.g.  If  then 
  • These allow you to differentiate linear terms in  and constants
• Funcitons involving roots will need to be rewritten as fractional powers of  first
  • e.g.  If  then rewrite as and differentiate
• Functions involving fractions with denominators in terms of  will need to be rewritten as negative powers of  first
  • e.g.  If then rewrite as and differentiate

How do I differentiate sums and differences of powers of x?

•  The formulae for differentiating powers of  apply to all rational powers so it is possible to differentiate any expression that is a sum or difference of powers of 
  • e.g.  If then
    
    
• Products and quotients cannot be differentiated in this way so would need expanding/simplifying first
  • e.g.  If then expand to which is a sum/difference of powers of  and can be differentiated

What might I be asked to do once I’ve found the derivative (gradient function)?

• The derivative can be used to find the gradient of a function at any point
  • The gradient of a function at a point is equal to the gradient of the tangent to the curve at that point
  • A question may refer to the gradient of the tangent
  • A GDC can be used to help find gradients or verify answers
• Take extra care when differentiating negative and fractional powers of 
  •  but