Differentiation is the biggest topic in the Additional Math papers. It appears as straightforward derivatives, stationary points, connected rates of change, small changes, and optimisation. The reports show that most students know the rules. The marks are lost in the details of the rules, the brackets, and finishing the question. This guide collects what Cambridge examiners say students get wrong, based on recent reports.
Summarised in our own words from a reading of the examiner reports.
Mistakes that cost marks
- Derivative errors. Not differentiating ln 3x correctly, putting the terms in the numerator of the quotient rule the wrong way round, and giving the derivative of tan(x/2) as ½ sec(x/2) instead of ½ sec²(x/2). Others used the quotient rule when a standard result should have been known, or used the product rule when it complicated the working.
- Brackets and signs. The reports say bracketing matters: many candidates obtained incorrect results from incorrect or missing brackets. Sign errors were common, particularly in the final term.
- Combining fractions. Sign or arithmetic errors, more common when the lowest common multiple of the denominators was not used.
- Small changes. Many candidates were less well prepared for the small changes relationship and just calculated the derivative at the point.
- Substituting a value. Omitting to substitute a value that should have been shown (such as x = −π/4), having the calculator in degree mode, and using a wrong value of h. The reports say the substitution of the value into the derivative must be shown, in case the value or the derivative is wrong.
- Connected rates of change. Using the chain rule is the standard method (for example dV/dt = 9 and dV/dr = 4πr²). Errors included confusing x with r, rounding r and then not finding the corresponding value of V, offering both positive and negative values of r without always rejecting the negative one, and slips when rearranging 4πr² = 9.
- Stationary points and optimisation. Finding the correct value of r or x and not going on to find the value that was asked for (the minimum surface area, or S), or thinking that the value of the second derivative at that point is the minimum value. Others substituted their value into the second derivative and obtained an incorrect or incorrectly rounded value.
- Setting up the expression. Errors in the expression for the area or volume (such as the area of a triangle), a slip when miscopying their own work, and stating the distance AC as 300 + x when AC² was 300² + x².
- "Show that." Many omitted the required step of showing eˣ as a factor, or of showing the division by eˣ, and moved directly from the expanded numerator to the given answer.
- Exact form. The reports suggest that centres ensure candidates understand the word "exact", and that final answers are in the correct form. For example, leaving −ln(1/2) and not simplifying to ln 2, and simplifying (−4 ± √32)/2 as −2 ± √32, which is incorrect.
- Domain. Giving both the positive and negative solutions inside a logarithm, not recognising that only the valid value should be used.
- Checking. Values such as a = −29 and b = −513 should alert candidates to an error, especially when the cubic that follows does not factorise.
What examiners advise
- Use brackets consistently in every statement. The reports say this maintains the correct structure of the steps.
- Show the substitution of the value into the derivative.
- Finish the question: after finding r or x, find the quantity that the question asks for, and decide whether it is a maximum or a minimum with a clear test.
- Give final answers in the correct form, exact where required.
- Check values for sense, especially when the numbers are unusually large.
- Know the standard derivatives, and choose the simplest rule.
Habits that help
- Write the chain rule for connected rates as a line of three fractions before you substitute anything.
- For optimisation, write "we want ..." (for example "the minimum surface area") at the top of the page.
- After you find a stationary point, substitute it back into the original function.
Questions students ask
Why are brackets so important in Additional Math differentiation?
Many candidates obtained incorrect results from incorrect or missing brackets, and sign errors were common. Use brackets consistently in every statement.
How do I use small changes?
Use the relationship δy ≈ (dy/dx)δx. Many candidates were less well prepared for small changes and just calculated the derivative at the point.
How do I solve connected rates of change questions?
Use the chain rule, for example dV/dt = dV/dr × dr/dt with dV/dr = 4πr². Errors included confusing x with r and rounding r before using it again.
What do I do after finding r or x in an optimisation problem?
Find the quantity the question asks for, such as the minimum surface area, and decide whether it is a maximum or a minimum with a clear test.
What does "exact" mean in differentiation answers?
A form such as an expression in ln or π, not a decimal. Leaving −ln(1/2) unsimplified lost marks, and giving both positive and negative solutions inside a logarithm ignores the domain.
How should I handle a "show that" question?
Show the required steps, such as eˣ as a factor and the division by eˣ. Many moved directly from the expanded numerator to the given answer.
Practice this topic
Ready to try some? Work through past paper questions on this topic one at a time. Attempt each question fully first, and open the mark scheme only afterwards.