SC06 Calculus: The Six Places Students Lose Marks

The single most common slip in differentiation. Asked for the derivative of y = (3x² + 1)⁵, many students write 5(3x² + 1)⁴ and stop — dropping the inner 6x. The correct answer is 5(3x² + 1)⁴ × 6x = 30x(3x² + 1)⁴. A habit that catches it: whenever the bracket is not just a bare x, multiply by the derivative of the inside after differentiating the outside. Same for trig — d/dx[sin(2x)] is 2cos(2x), not cos(2x).

The quotient rule is (u/v)′ = (u′v − uv′) / v². Marks are lost almost entirely on that middle minus sign — written as a plus, or with u′v and uv′ swapped. A phrase that helps: "bottom times derivative of top, minus top times derivative of bottom, all over bottom squared." Dropping the + C on an indefinite integral is the other guaranteed loss. ∫2x dx = x² + C; without the + C that is a mark gone in the UEC, however complete the rest of the working. Write the + C the moment you write the integral sign.

For ∫ from a to b, it is "value at the upper limit minus value at the lower limit," never the other way. ∫ from 1 to 3 of 2x dx = [x²] from 1 to 3 = 9 − 1 = 8, not 1 − 9. When finding an area and the curve dips below the x-axis, the definite integral comes out negative — take the absolute value for area. If the curve crosses the x-axis within the interval, split the integral at the crossing; integrating straight through and taking the modulus at the end lets the positive and negative parts cancel.

Solving f′(x) = 0 only locates the stationary points — it does not say whether each is a maximum, a minimum or a point of inflexion. UEC structured questions almost always ask for the classification, so "stationary point at x = 2" alone leaves marks on the table. Two tests: the second-derivative test — f″(x) > 0 means a minimum, f″(x) < 0 a maximum, and f″(x) = 0 means you must fall back on the sign of f′(x) either side; or check the sign change of f′(x) directly across the point. The second-derivative test is usually faster in an exam, but when f″(x) = 0 you have to switch methods rather than guess.

These six — chain rule, quotient-rule signs, + C, integral limits, area sign, stationary-point classification — together account for most of the marks lost in calculus. What they share: it is not that you cannot do the question, it is that the check was incomplete. Next time you do a calculus question, run this list before you move on. DuckMath's grading locates the step where your working first drifts; do enough and you will see you keep falling into the same one or two pits — those are the ones to watch.

Now go practise

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