Playbook
Applications of Derivative
183 q - 3.81/paper - 23% HARD. The cheapest cornerstone by some distance, and the best marks-per-hour chapter on the paper. Seven subtopics, none above 31% HARD; Approximations using Differentials is 11 q at 0% HARD.
- Questions in the bank
- 183
- q/paper (2024–25 shifts)
- 3.81
- Tagged HARD
- 23%
- Subtopics
- 7
Strand: Cornerstone
When you’ll see it
A derivative used for something rather than computed: a maximum, a rate, a tangent slope, an interval of increase, an approximation, or a mean-value statement.
How this chapter is tested
183 q at 3.81 per paper and only 23% HARD. This is the cheapest cornerstone by some distance and, measured as marks per hour of preparation, the best chapter on the paper. Seven subtopics and not one of them is above 31% HARD — there is no expensive corner to be afraid of here, which is not true of any other cornerstone.
Approximations using Differentials is the extreme case: 11 q and 0% HARD across the whole bank. Angle Between Curves and Orthogonality is 8 q at 13%, and Rolle's Theorem and Mean Value Theorem is 18 q at 17%. That is 37 questions of nearly-free content sitting inside a cornerstone, which is an unusual thing to find and worth taking early.
The volume, though, sits in Maxima, Minima and Optimisation (42 q) and Rate of Change and Related Rates (40 q). Both are word problems, and the difficulty is almost never the calculus — it is setting up the right function or the right chain of dependencies before differentiating. Drill the setup, not the differentiation; the differentiation is Differentiation's job.
One efficiency note that pays at 1.8 minutes a question: optimisation questions in this chapter frequently have a geometric shortcut, and the survey found the same shortcut in other chapters too. A maximum distance from a point to a circle is centre distance plus radius; no calculus is required. Recognising when the calculus can be skipped is a time lever, not a stylistic preference.
The sub-skills
The distinct skills inside the chapter, in the order to learn them.
Approximations using differentials
A small change in x produces roughly the derivative times that change in y. 11 q at 0% HARD — the only subtopic in the subject that has never produced a HARD question apart from Linear Programming's objective function. Learn it in one sitting.
Tangents, normals, and the slope of a curve
The derivative at a point is the tangent slope; the normal slope is its negative reciprocal. 35 q at 29% HARD. Everything later in the chapter reads slopes off this skill.
Angle between curves and orthogonality
Find the intersection point, take both tangent slopes, then apply the two-line angle formula. Orthogonal curves are the product of slopes equal to minus one — the same perpendicularity condition Straight Line uses. 8 q at 13% HARD.
Rate of change and related rates
Differentiate a geometric relation with respect to time and substitute the instantaneous values LAST. 40 q at 20% HARD, and the second-largest block in the chapter.
Increasing and decreasing functions
Sign of the first derivative on an interval. The work is solving an inequality cleanly and reporting the interval in the form the options use. 29 q at 31% HARD.
Maxima, minima, and optimisation
Build the objective, reduce it to one variable using the constraint, then differentiate. 42 q at 29% HARD, the largest subtopic. The reduction step is where the marks are won and lost.
Rolle's theorem and mean value theorem
Three hypotheses to check before any conclusion: continuity on the closed interval, differentiability on the open one, and for Rolle equal endpoint values. 18 q at 17% HARD, and mostly hypothesis-checking rather than computation.
Traps to expect
Distractor shapes this chapter reuses. The Traps page covers the patterns that cut across chapters.
Conclusion applied where a hypothesis fails
A Rolle or MVT question set on a function with a corner or a discontinuity in the interval. The trap option reports the value the theorem would give if it applied. Check the three hypotheses first, every time.
Tangent slope offered where the normal was asked
The derivative value and its negative reciprocal both appear in the option list. Read the question word before you compute.
Substituting the instant before differentiating
In a related-rates problem, putting the numerical values in first freezes the variable and kills the rate. Differentiate the general relation, then substitute.
Local extremum reported as global
On a closed interval the answer can sit at an endpoint where the derivative never vanishes. The critical-point value is the distractor.
Learn it before you drill it
This chapter has full teaching notes — foundations, worked examples, self-checks and a per-subtopic mastery checkpoint. Read the notes once, then drill subtopic by subtopic below.
Applications of Derivative notesDrill every applications of derivative question
183 questions from the bank, scoped to 7 bundled subtopics.
Drill one subtopic at a time
The 7 subtopics this playbook covers, in catalog order.
Related playbooks
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