CIE IGCSE Maths

Revision Notes

2.16.1 Differentiation - Basics

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 – ie. its speed
    • The rate at which a virus spreads amongst a population

 

Diff Basics Notes fig1, downloadable IGCSE & GCSE Maths revision notes

 

  • To begin to understand differentiation you’ll need to understand gradient (see Finding Gradients of Non-Linear Graphs)

Gradient

  • Gradient generally means steepness.
    • For example, the gradient of a road up the side of a hill is important to lorry drivers

 

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

  • 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 is a way of describing the rate at which change happens

Straight lines and curves

  • For a straight line the gradient is always the same (constant)
    • Recall y= mx + c, where m is the gradient (see Straight Lines – Finding Equations)

 

Diff Basics Notes fig3, downloadable IGCSE & GCSE Maths revision notes

  • 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

Diff Basics Notes fig4, downloadable IGCSE & GCSE Maths revision notes

 

  • The derived function (aka gradient function) is an expression that allows the gradient to be calculated anywhere along a curve
  • The derived function is also called the derivative

How do I find the derived function or derivative? 

  • This is really where the fun with differentiation begins!
  • The derived function (dy/dx) is found by differentiating y

 

Diff Basics Notes fig5, downloadable IGCSE & GCSE Maths revision notes

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

STEP 1   Multiply by the power

STEP 2   Take one off the power

 

Diff Basics Notes fig6, downloadable IGCSE & GCSE Maths revision notes

How do I find the value of a gradient?

  • Substitute the x value into the expression for the derived function and evaluate it

 

Diff Basics Notes fig7, downloadable IGCSE & GCSE Maths revision notes

Exam Tip

When differentiating long, awkward expressions, write each step out fully and simplify afterwards.

Worked Example

Diff Basics Example fig1, downloadable IGCSE & GCSE Maths revision notes

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