Differentiation in A Level Math Paper 1: common mistakes and examiner advice

Differentiation questions in Paper 1 are rarely about the rules themselves. Most students can differentiate a polynomial. Marks are lost in the chain rule, in what is set equal to zero, and in how the answer is used in the rest of 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

• Not applying the chain rule. This is the most common error in the calculus reports. The derivative of something like (3x − 2)^½ or (2x + 1)⁻¹ loses a factor, usually the one from inside the bracket. It also appears when finding a second derivative, and in rates of change.
• Setting the wrong thing equal to zero. At a stationary point, dy/dx = 0. Students set y = 0 instead, or used another given value of dy/dx in place of zero, or equated the second derivative to zero.
• Only finding the x-coordinate. A stationary point has two coordinates. Several reports mention students who found only x and forgot the y-value.
• Power and sign errors. Incorrect powers when differentiating negative or fractional powers, and arithmetic slips such as (−2)³ evaluated as 8. Some students also left out a term, such as −x, when copying the function from the question.
• Mixing up tangent and normal. Using the value of dy/dx as the gradient of the normal, or using a given point where the equation of the line needed a different one. The normal's gradient is the negative reciprocal of the tangent's.
• Constants of integration. When dy/dx is given and the curve is wanted, the constant is omitted or not evaluated. The reports describe this as a very frequent slip.
• Second derivative. Not calculating the value of the second derivative, or drawing the wrong conclusion about the nature of the stationary point.
• Rates of change and connected rates. Substituting a value for the variable into the wrong formula, such as using h = 10 in a volume formula before differentiating, or not forming the chain rule equation dV/dt = dV/dh × dh/dt.
• Increasing and decreasing functions. Stating that a derivative is positive without justification. Some thought the derivative could be zero within the condition for increasing. Others did not link the range of values where the derivative is positive to the given domain.

What examiners advise

• Think about what the question requires before you differentiate. Weaker candidates, one report says, decided at once that a given gradient had to be integrated, without thinking about what was being asked.
• Show both coordinates for a stationary point, and say how you know its nature.
• Show the substitution of values into your derivative. Examiners could not give credit when only the final answer was written.
• Check against the given diagram. If a limit or a value does not fit the diagram, something is wrong.
• Use the information from earlier parts. A mistake in part (a) that is used correctly later still earns marks, but if it leads to an equation that cannot be solved, go back and check.

Habits that help

• Write the function in power form before differentiating.
• For the chain rule, write "inside" and "outside" before you start.
• Put brackets around negative values when substituting.

Questions students ask

Why do I lose marks on the chain rule?

The derivative of something like (3x − 2)^½ or (2x + 1)⁻¹ loses a factor, usually the one from inside the bracket. Examiners call it the most common error in the calculus reports. Write "inside" and "outside" before you start.

What do I set equal to zero at a stationary point?

dy/dx, not y. Students also set the second derivative to zero or use another given value of dy/dx in place of zero.

Do I need both coordinates of a stationary point?

Yes. A stationary point has x and y values, and several reports mention students who found only x. Also say how you know its nature, using the second derivative.

What is the gradient of the normal?

The negative reciprocal of the gradient of the tangent. A recorded mistake is using the value of dy/dx as the gradient of the normal.

How do I find the equation of a curve from its gradient?

Integrate and include the constant, then find the constant using a given point. The reports describe a forgotten constant as a very frequent slip.

How do I show that a function is increasing?

Show that the derivative is positive and give a reason, and link the range where it is positive to the domain given. Reports record students stating that a derivative is positive without justification.