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Calculus - Level 3 Differentiation

Immediately below this paragraph is a table with links to the exam questions and answers for all the externals way back to 2013. Focus on the exams for the last three years. Recognise the type of question e.g This is a question about..... Know how you are going to set out your working when answering each type of question.

Year	Differentiation

2025	Questions	Answers
2024	Questions	Answers
2023	Questions	Answers
2022	Questions	Answers
2021	Questions	Answers
2020	Questions	Answers
2019	Questions	Answers
2018	Questions	Answers
2017	Questions	Answers
2016	Questions	Answers
2015	Questions	Answers
2014	Questions	Answers
2013	Questions	Answers

Differentiation

We use the Walker maths revision book. This external standard is worth 6 credits.

• READ the question, all of it. There will be useful information
• ANSWER the question. If it asks for the equation of a curve, then give the equation, not just the derivative or slope of the gradient line. If it asks for the co-ordinates of a point, give the y-value as well as the x-value

The derivative of a curve/function at a point is the slope of the tangent line to the curve at that point. The derivative is sometimes called the 'rate of change'.

• When there is a square root symbol, you will convert this to an exponent before you differentiate
• An interval can be described as -2 < x ≤ 1 OR (-2, 1]

Product Rule, Quotient Rule, Composite functions

• You guys all know how to use these rules. Well done
• First, write the rule down
• Then, write down f = .... and g = ...
• Underneath these write f' = .... and g' = ...
• Then substitute f, g, f', and g' into the formula
• remember to multiply by the derivative of the composite part of the function (if there is a composite part)

• The chain rule is covered on pages 30-37 of Walker, together with exponential functions, log functions and trig functions
• The product rule is covered on pages 40-44 of Walker
• The quotient rule is covered on pages 45-47 of Walker

 
Questions 1a and 2a from the 2022 exam

Maxima, Minima, Points of Inflection

Stationary points are covered on pages 61-72 of Walker
 

• the derivative is zero at stationary points
• if the second derivative is positive the point is a minimum
• if the second derivative is zero the point is a point of inflection
• if the second derivative is negative the point is a maximum

 
Question 1b from the 2022 exam

Equation of a normal

Tangents and normals are covered on pages 51-60 of Walker.
 

• The slope of the tangent times the slope of the normal = -1

 
Question 1c from the 2022 exam

Kinematic Equations

Kinematics are covered on pages 86-89 of Walker.
 

• The derivative of distance is velocity, v(t) = ds/dt
• The derivative of velocity is acceleration, a(t) = dv/dt
• When an object is stationary its velocity is zero

 
Question 2c from the 2022 exam

Optimisation Problems

Question 3e from the 2022 exam

Related Rates of Change

Related Rates of Change are covered on pages 73-85 of Walker.
 

• In these problems you are asked to find, say, dh/dt, and you are given a related rate of change such as dV/dt, and a formula connecting 'V' and 'h'
• The technique is to write dh/dt as dh/dV times dV/dt, and to remember that dh/dV is just the reciprocal of dV/dh

 
Question 3c from the 2022 exam

Derivatives of parametric functions

Parametric functions are covered on pages 48-50 of Walker.
 

• Both 'x' and 'y' are expressed as functions of 't'
• calculate dy/dt and dx/dt
• then dy/dx = dy/dt times dt/dx

 
Question 1d from the 2022 exam

Properties of Graphs

These are covered on pages 16-27 of Walker.
 

• A curve is continuous if it can be drawn without lifting the pen from the paper
• A curve is smooth if just one tangent can be drawn at every point on the curve
• A curve is differentiable at a point if is continuous and smooth at that point
• The limit exists at a point if the curve approaches the same value from both sides of the point
• A curve is concave in an interval if it has an upright cup shape (curves inward)
• A curve is convex in an interval if it has an upside down cup shape (curves outward)

 
Question 3b from the 2022 exam