Introduction to Derivatives (Edexcel IGCSE Further Pure Maths)

Revision Note

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Introduction to Derivatives

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

Example of measuring the speed of a car

 

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

  • 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 with respect to x'

  • So gradient describes the rate at which change happens

Diff Basics Notes fig2, downloadable IGCSE & GCSE Maths revision notes

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

Gradient of a line
  • 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

Gradient of a curve

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

  • The equation of a curve can be given in the form y equals straight f open parentheses x close parentheses

    • 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 fraction numerator straight d y over denominator straight d x end fraction

    • This is pronounced "dy by dx" or "dy over dx"

    • In function notation, it can be written straight f apostrophe open parentheses x close parentheses

      • pronounced "f-dash-of-x" or "f-prime-of-x"

  • To get from y equals straight f open parentheses x close parentheses to fraction numerator straight d y over denominator straight d x end fraction equals straight f apostrophe open parentheses x close parentheses 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 fraction numerator straight d y over denominator straight d x end fraction equals straight f to the power of apostrophe open parentheses x close parentheses for a curve, you can find the gradient for any point on the curve

    • straight f to the power of apostrophe open parentheses a close parentheses gives the value of the gradient when x equals a

      • See the 'Finding Gradients' revision note for more details

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Mark Curtis

Author: Mark Curtis

Expertise: Maths

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.

Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

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.