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Differentiation

Simplify before differentiating, and count the points where a derivative fails to exist. Lighter than in the early papers.

Questions in the bank
77
q/paper in 2025–26
0.35
Numeric answer
31%
Notes pages
5

Tier: Long tail

When you’ll see it

The derivative of a composite, implicit, parametric or inverse function, or whether a function is differentiable at a point.

How this chapter is tested

The chapter has two halves. One computes a derivative, and the work is almost always in the simplification before it: an inverse-trig expression that collapses to a multiple of tan⁻¹ x, a power that is easier after taking logs, a curve given by a parameter. The other asks whether a derivative exists: where two pieces join, where a modulus, a max or a min has a corner, and where the greatest integer function jumps.

The computing half is fast when the simplification is seen and slow when it is not; the chain rule applied blindly works but costs minutes. The existence half is a count with traps: a zero inside a modulus is only a candidate, a jump is both a discontinuity and a point of non-differentiability, and continuity must hold before slopes are matched.

Functional equations such as f(x + y) = f(x) f(y) sit here too, solved by fixing f(0) first. The chapter lies between Limits and Continuity, which supplies the definitions, and Application of Derivatives, which uses every rule.

The sub-skills

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

  • Chain rule and inverse-trig simplification

    Simplify by x = tan θ or a similar substitution first; for g = f⁻¹, g′(k) = 1/f′(a) where f(a) = k.

  • Implicit, parametric and logarithmic differentiation

    dy/dx = (dy/dt)/(dx/dt); d²y/dx² is the t-derivative of dy/dx divided by dx/dt; take logs of powers like xˣ.

  • Functional equations and derivative constants

    Fix f(0) from the relation first; treat f′(1) inside a polynomial as a number, not a function.

  • Differentiability of piecewise functions

    Continuity at the join, then equal left and right derivatives; at a single point, use the limit that defines f′.

  • Counting non-differentiable points

    List the zeros inside each modulus, the crossings of a max or min, and the jumps of [x], then test each one.

Traps to expect

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

  • Not the ratio of second derivatives

    For x = t², y = t³, the ratio of second derivatives gives 3t, but d²y/dx² = 3/(4t).

  • The branch decides the answer

    cos⁻¹(cos x) = x only on [0, π]; on [π, 2π] it is 2π − x, with derivative −1.

  • Matching slopes is not enough

    Equal one-sided slopes at a jump do not make f differentiable; check continuity first.

  • A jump counts twice

    When a question adds the points of discontinuity and of non-differentiability, each jump belongs to both counts.

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.

Differentiation notes

Drill every Differentiation question

77 questions from the bank, across 5 subtopics.

Drill the 77 questions

Drill one subtopic at a time

The 5 subtopics, in teaching order.

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