Definition of Derivatives (CIE A Level Maths: Pure 1)

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Definition of Derivatives

What is the derivative of a function? 

  • Differentiation is an operation that calculates the rate of change of a function with respect to a variable
    • This means how much the function varies when the variable increases by one unit
  • To differentiate a function (f(x)) with respect to the variable x we use the notation
    • fraction numerator straight d over denominator straight d x end fraction open parentheses straight f open parentheses x close parentheses close parentheses
  • The result is called the derivative

What is the link between derivatives and gradients? 

  • The rate of change of a function f(x) with respect to x can be thought of as the gradient function of the graph y = f(x)
    • We can write the gradient function (or derivative) as
      • straight f apostrophe left parenthesis x right parenthesis or fraction numerator d y over denominator d x end fraction  
  • The rate of change of a function (or the gradient of its graph) varies for different values of x
    • For a linear function f(x) = mx + c the gradient is constant
      • We can write this as straight f apostrophe left parenthesis x right parenthesis equals m or fraction numerator d y over denominator d x end fraction equals m
    • For the quadratic function f(x) = x² the gradient varies
      • Near the origin the gradient is close to 0
      • As x increases the gradient of the graph increases

How can I find the derivative of a function at a point? 

  • The derivative of a function (or gradient of its graph) at a point is equal to the gradient of the tangent to the graph at that point
  • To estimate the gradient you could draw the tangent and calculate its gradient
  • To find the actual gradient at a point x
    • Pick a second point on the curve close to the first point (call it x + h)
    • Calculate the gradient of the chord joining the two points
      • fraction numerator straight f open parentheses x plus h close parentheses minus straight f left parenthesis x right parenthesis over denominator open parentheses x plus h close parentheses minus x end fraction equals fraction numerator straight f open parentheses x plus h close parentheses minus straight f left parenthesis x right parenthesis over denominator h end fraction
    • Move the second point closer to the first point (make h get close to zero)
    • Examine what happens to the gradient of the chord
    • The gradient of the tangent will be the limit of the gradients of the chords
      • straight f to the power of apostrophe open parentheses x close parentheses equals limit as h rightwards arrow 0 of invisible function application fraction numerator straight f open parentheses x plus h close parentheses minus straight f left parenthesis x right parenthesis over denominator h end fraction
      • You do not need to remember this formula

5-1-2-definiton-of-derivatives-diagram-1

Worked example

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Examiner Tip

  • Deriving a derivative from scratch is not examinable
  • This revision note is intended to give you an understanding of what derivatives do

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Dan

Author: Dan

Expertise: Maths

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.