A-Level Differentiation: The 8 Mistakes That Cost Marks
Most students lose differentiation marks in the same handful of ways. Not because the topic is hard, but because a single slip early in the working carries all the way to the final line.
What follows is the eight we see most often. For each one there is the wrong answer, the right answer, and a way to catch yourself doing it. Work through them and you will recognise your own habits fairly quickly — most students have two or three they make repeatedly rather than eight they make occasionally.
1. Multiplying by the index but forgetting to reduce it
This is the single most common error in the whole topic, and it usually appears under time pressure rather than through not knowing the rule.
The power rule has two steps: multiply by the index, then reduce the index by one. Miss either step and the answer is wrong.
Differentiate y = 4x³.
✗ 12x³ — multiplied by 3, forgot to reduce the power
✗ 4x² — reduced the power, forgot to multiply
✓ 12x²
How to catch it: the power in your answer should always be one lower than the power you started with. Glance at the indices before and after. If a cubic term is still cubed, you have only done half the rule.
2. Differentiating a constant into itself
A constant term has no x in it, so it cannot change as x changes — its gradient is zero. It disappears entirely.
Differentiate y = 5x² − 7.
✗ 10x − 7 — the constant has been carried through untouched
✓ 10x
How to catch it: write the constant as −7x⁰ if it helps. The power rule then gives 0 × −7x⁻¹ = 0, and it is obvious why it vanishes.
3. Getting negative indices the wrong way round
Reducing an index by one always makes it more negative. Students who are comfortable with positive powers often reverse this without noticing.
Differentiate y = 3/x².
Rewrite first: y = 3x⁻².
✗ −6x⁻¹, or −6/x — the index moved the wrong way
✓ −6x⁻³, or −6/x³
How to catch it: −2 − 1 = −3, not −1. Say the subtraction out loud if you have to. The sign of the coefficient flips too, because you multiplied by a negative index.
4. Substituting the value before you differentiate
When a question asks for the gradient at a particular point, the order matters absolutely.
Find the gradient of y = x² − 4x + 1 at x = 3.
✗ Substitute first: y(3) = −2, a constant, whose derivative is 0
✓ Differentiate first: dy/dx = 2x − 4, then substitute: 2(3) − 4 = 2
How to catch it: if you have substituted and are left with a number, stop — you have found a point on the curve, not a gradient. Differentiation has to happen while x is still a variable.
5. Using the tangent gradient when the question asked for the normal
The normal is perpendicular to the tangent, so its gradient is −1 divided by the tangent gradient. Two things go wrong here: using the tangent gradient unchanged, or taking the reciprocal without the minus sign.
Find the equation of the normal to y = x² at x = 3.
The tangent gradient is dy/dx = 2x = 6, and the point is (3, 9).
✗ y = 6x − 9 — that is the tangent, not the normal
✗ Normal gradient 1/6 — the reciprocal, but the minus sign is missing
✓ Normal gradient −1/6, giving y = −x/6 + 19/2
How to catch it: multiply your two gradients together. If they do not give −1, they are not perpendicular. Here 6 × (−1/6) = −1. ✓
6. Forgetting the inside derivative in the chain rule
The chain rule is: differentiate the outside, then multiply by the derivative of the inside. The second half is the part that gets dropped.
Differentiate y = (3x² + 1)⁴.
✗ 4(3x² + 1)³ — the outside has been differentiated, the inside ignored
✓ 24x(3x² + 1)³ — because the derivative of the inside is 6x, and 4 × 6x = 24x
How to catch it: if the bracket contains anything other than plain x, there must be an extra factor in front of your answer. No new factor appearing is a reliable sign the inside derivative has been forgotten.
7. Differentiating both factors in the product rule
When two functions are multiplied, you cannot differentiate each one and multiply the results. The product rule exists precisely because that does not work.
Differentiate y = x² sin x.
✗ 2x cos x — each factor differentiated separately
✓ x² cos x + 2x sin x
With u = x² and v = sin x, the rule gives u(dv/dx) + v(du/dx), so you always end up with two terms added together.
How to catch it: a product rule answer has two terms. If yours has one, you have almost certainly differentiated both factors and multiplied.
8. Reversing the order in the quotient rule
The quotient rule is not symmetric. Swapping the two terms on the numerator flips the sign of the entire answer.
For y = u/v, the rule is:
dy/dx = [ v(du/dx) − u(dv/dx) ] ÷ v²
In words: bottom times the derivative of the top, minus top times the derivative of the bottom.
Differentiate y = (sin x)/x².
✓ (x cos x − 2 sin x) / x³
✗ The same expression with the numerator reversed — every sign is wrong
How to catch it: the term that starts the numerator is the one containing the derivative of the top. And before reaching for the rule at all, check whether you can simply divide: something like (4x³ − 6x)/x² is far quicker as 4x − 6x⁻¹.
Bonus: stopping after one differentiation
When a question asks for d²y/dx², or asks you to classify a stationary point, you have to differentiate twice.
For y = 2x⁴ − 5x²:
First derivative: 8x³ − 10x
Second derivative: 24x² − 10
Classifying a stationary point uses the sign of the second derivative: positive means a minimum, negative means a maximum.
For y = x³ − 3x, the stationary points are at x = ±1, and d²y/dx² = 6x. At x = −1 that is −6, so it is a maximum. At x = 1 it is 6, so it is a minimum.
How to find out which mistakes you make
Reading a list like this is useful, but it will not tell you which of these you personally keep doing. That takes practice with feedback, and specifically feedback that tells you what went wrong rather than just marking the question wrong.
Our A-Level differentiation revision engine generates a new question every time and marks your answer instantly. When you get one wrong it identifies the error — whether you forgot to reduce the index, used the tangent gradient instead of the normal, or stopped after one differentiation — so the pattern in your own working becomes visible. It covers all thirteen differentiation topics, from first principles through to implicit and parametric differentiation, and it is free to use.
If you would rather work through it with someone, you can also book a free lesson with an eClassroom maths tutor.
Quick reference: the eight mistakes
Set this as a bulleted list. Wix Blog tables are awkward on mobile, so a list is safer.
Multiplied but did not reduce the index — is the power one lower than you started with?
Kept the constant term — a number on its own always differentiates to zero
Moved a negative index the wrong way — −2 − 1 = −3, not −1
Substituted before differentiating — if you have a number, you found a point, not a gradient
Used the tangent gradient for the normal — the two gradients must multiply to −1
Forgot the inside derivative in the chain rule — a non-trivial bracket must produce a new factor
Differentiated both factors in the product rule — the answer should have two terms
Reversed the quotient rule numerator — bottom × derivative of top comes first