Edexcel A Level Maths: Pure

Revision Notes

9.2.1 Parametric Differentiation

A Level Only

How do I find dy/dx from parametric equations? 

  • Ensure you are familiar with Parametric Equations – Basics first

 Notes para_diff, AS & A Level Maths revision notes


  • This method uses the chain rule and the reciprocal property of derivatives
    • dy/dx = dy/dt × dt/dx
    • dt/dx = 1 ÷ dx/dt
  • Equivalently, dy/dx = dy/dt ÷ dx/dt
  • dy/dx will be in terms of t – this is fine
    • Questions usually involve finding gradients, tangents and normals
  • Chain rule is always needed when there are three variables or more – see Connected Rates of Change


How do I find gradients, tangents and normals from parametric equations? 

  • To find a gradient …

 Notes para_grad, AS & A Level Maths revision notes

  • STEP 1: Find dx/dt and dy/dt
  • STEP 2: Find dy/dx in terms of t
    Using either dy/dx = dy/dt ÷ dx/dt
    or dy/dx = dy/dt × dt/dx where dt/dx = 1 ÷ dx/dt
  • STEP 3: Find the value of t at the required point
  • STEP 4: Substitute this value of t into dy/dx to find the gradient
  • … to then go on to find the equation of a tangent …

Notes para_tan, AS & A Level Maths revision notes

  • STEP 7: Use perpendicular lines property to find the gradient of the normal
    m1 × m2 = -1
  • STEP 8: Use gradient and point to find the equation of the normal
    y – y1 = m(x – x1)


What else may I be asked to do?

  • Questions may require use of tangents and normals as per the coordiante geometry sections
    • Find points of intersection between a tangent/normal and x/y axes
    • Find areas of basic shapes enclosed by axes and/or tangents/normal
  • Find stationary points (dy/dx = 0)

Notes para_stat, AS & A Level Maths revision notes

  • You may also be asked about horizontal and vertical tangents
    • At horizontal (parallel to the x-axis) tangents, dy/dt = 0
    • At vertical (parallel to y-axis) tangents, dx/dt = 0


Notes para_hor_ver_tang, AS & A Level Maths revision notes

Just for fun …

  • Try plotting the graph from the question below using graphing software
  • Plenty of free online tools do this – for example Desmos and Geogebra
  • Try changing the domain of t to π/3 ≤ t ≤ π/3
A Level Only

Worked Example

Example soltn_a, A Level & AS Level Pure Maths Revision NotesExample soltn_b, AS & A Level Maths revision notes


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