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Application of Derivatives

Maxima and minima, monotonicity and tangents. It has shrunk by about half since the early papers, but still appears on many papers.

Questions in the bank
140
q/paper in 2025–26
0.59
Numeric answer
27%
Notes pages
8

Tier: Long tail

When you’ll see it

A slope or a rate, where a function rises or falls, how many real roots an equation has, or the greatest or least value of something.

How this chapter is tested

The chapter opens with the derivative as a slope and a rate — tangents, normals, curves meeting at an angle, related rates — and then uses its sign. The largest group of questions asks for a maximum or a minimum: local ones from sign changes of f′, the greatest and least values on an interval, and word problems.

The routine questions are one derivative and one sign chart. Time goes where the sign chart hides a trap: a double zero of f′ that is not an extremum, a corner where f′ does not exist, an endpoint higher than every local maximum, or a parameter that must keep f′ ≥ 0 for every x. Working backwards, from where a cubic's extrema sit to its coefficients, also takes longer.

Every derivative comes from Differentiation. When f′ is a quadratic that must keep one sign, the condition is a discriminant from Quadratic Equations, and tangents and shortest distances to curves lean on Straight Lines and Conic Sections.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Tangents, normals and rates

    Tangent slope f′(a), normal slope −1/f′(a); differentiate a relation in time before putting in the instant's values.

  • Increasing and decreasing functions

    Read the sign of f′; for f increasing on all of ℝ with a quadratic f′, need a positive leading coefficient and discriminant ≤ 0.

  • Counting real roots

    A strictly monotonic function has at most one root; otherwise the signs of the local maximum and minimum values decide.

  • Local maxima and minima

    An extremum needs f′ to change sign; check corners and cusps where f′ does not exist.

  • Functions built from their extrema

    Each extremum gives f′ = 0 there; a limit or a given value fixes the remaining coefficients.

  • Greatest and least values, and optimisation

    On a closed interval compare critical values with the endpoint values; in a word problem reduce to one variable and stay inside its allowed range.

  • Rolle's and mean value theorems

    Equal end values force a zero of f′; k distinct zeros of f give at least k − 1 zeros of f′.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • Even powers do not change sign

    A factor like (x − a)² in f′ gives a critical point that is not a maximum or a minimum.

  • The endpoints count

    On a closed interval the greatest or least value can sit at an endpoint, beyond every local extremum.

  • Normal, not tangent

    A normal parallel to a line of slope m needs f′ = −1/m. Solving f′ = m finds the tangent instead.

  • Each piece is not the union

    1/x decreases on x < 0 and on x > 0, but not on their union. Options often differ only in this.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Application of Derivatives notes

Drill every Application of Derivatives question

140 questions from the bank, across 8 subtopics.

Drill the 140 questions

Drill one subtopic at a time

The 8 subtopics, in teaching order.

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