Differentiation in A Level Math Paper 3: common mistakes and examiner advice

Paper 3 differentiation covers the product, quotient and chain rules, parametric and implicit differentiation, exponentials, logarithms and trigonometric functions, and uses of the derivative such as tangents, normals and stationary points. The reports say the rules are generally known. The marks are lost in the details of using them. 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

• The chain rule. This is the most repeated problem. Reports list the derivative of e^(x/2) being given as e^(x/2), 2e^(x/2) or eˣ, the statement d/dx (sin 3x) = cos 3x, and the chain rule not being applied to x or to y (or both) in a parametric question. A factor of 2 lost from the derivative of a function of 2x was also common.
• The exponential with a negative or fractional index. In e⁻ᵃˣ, the a was often incorrectly placed, and the sign was wrong. Reports also mention 3 appearing in place of 1/3 in e^((1−x)/3).
• The quotient rule. The denominator was omitted, the terms in the numerator were subtracted in the wrong order, and missing brackets in the numerator led to errors.
• The product rule. Students omitted eˣ when differentiating (eˣ − 1)⁻¹ by the product rule, or used a wrong sign in the formula.
• Logarithm derivatives. ln 5x was differentiated as 1/(5x) instead of 1/x, and ln(8 − x) was differentiated without the chain rule factor. Very few students who could not reach the given answer traced the error.
• Implicit differentiation. Students did not combine the derivatives of the two terms correctly, made sign errors, or misinterpreted "tangent parallel to the y-axis". That condition means the gradient is undefined, which corresponds to dx/dy = 0.
• The wrong gradient of the normal. Students left the gradient of the normal as the tangent gradient, or used the answer given in the question to find the answer given, giving 0 = 0.
• Taking ln when it was unnecessary. Several reports say that many students decided to take logarithms in a situation where it did not help.
• Dividing through by tan x and losing the solution x = 0. The same applies to dividing by any expression that could be zero.
• Inverse tangent. In differentiating tan⁻¹ 3x, the factor 3 was not always handled correctly, and common wrong answers included 3/(1 + 3x²) and 1/(1 + 9x²).
• Quoting instead of differentiating. Some students quoted a result from the formula book when the question asked them to differentiate the given function.
• Simplification. Many errors came after the differentiation: sign and transcription errors when rearranging, and incorrect cancelling.
• Notation. The notation y′ was often accurate but sometimes confused with y, and brackets were not used where needed: (…)² written as …².
• Radians. Working in degrees in a question that needed radians.
• Area must be positive. Only a few students recognised that an area has to be positive and so needs the modulus of the integral.

What examiners advise

• Show your working for each rule: write u, v, u′ and v′ before assembling a product or a quotient.
• Use brackets and watch the signs.
• When the answer is given, go back and check where the error is. Do not try to force the algebra.
• Do not divide by something that could be zero. Factorise instead.
• State the conclusion of any change-of-sign or tangent argument.
• Show the factorisation or the derivative on the right-hand side of an equation clearly.

Habits that help

• Differentiate in two stages for composite functions: the outside, then the inside.
• After finding a stationary point equation, check whether the factor you want to cancel could be zero.
• Substitute a simple value of x into the derivative as a check against your formula.

Questions students ask

Why do I keep losing marks on the chain rule in Paper 3?

It is the most repeated problem. Reports list the derivative of e^(x/2) given as e^(x/2) or eˣ, d/dx (sin 3x) given as cos 3x, and the chain rule not applied to x or y in a parametric question. Differentiate in two stages: the outside, then the inside.

What is the derivative of ln 5x?

1/x, not 1/(5x). Reports also record ln(8 − x) differentiated without the chain rule factor.

How do I use the quotient rule without mistakes?

Write u, v, u′ and v′ first. Common errors are leaving out the denominator, subtracting the numerator terms in the wrong order, and missing brackets in the numerator.

What does "tangent parallel to the y-axis" mean in implicit differentiation?

The gradient is undefined, which means dx/dy = 0. Reports record students misinterpreting this condition.

Why should I not divide through by tan x?

Because you can lose the solution x = 0. The same applies to dividing by any expression that could be zero. Factorise instead.

Is the area under a curve always positive?

An area has to be positive, so you need the modulus of the integral where the curve is below the axis. Only a few students recognised this.