First Principles Differentiation (AQA A Level Maths: Pure)

What is the derivative or gradient function?

• For a curve y = f(x) there is an associated function called the derivative or gradient function
• The derivative of f(x) is written as f'(x) or
• The derivative is a formula that can be used to find the gradient of y = f(x) at any point, by substituting the x coordinate of the point into the formula
• The process of finding the derivative of a function is called differentiation
• We differentiate a function to find its derivative 

What is differentiation from first principles?

• Differentiation from first principles uses the definition of the derivative of a function f(x)

• The definition is

•  means the 'limit as h tends to zero'
• When,  which is undefined
  • Instead we consider what happens as h gets closer and closer to zero
• Differentiation from first principles means using that definition to show what the derivative of a function is

How do I differentiate from first principles?

    so 

   

   

   

         

The derivative of  is