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LearnCambridge9231Current syllabusPaper 2 · Further Pure Mathematics 2Differentiation
Paper 2 · Module

Differentiation

2 subtopics≈ 90 min in total
Start the module: Inverse trig and hyperbolic functions in calculusOpen graphing calculator

  1. Inverse trig and hyperbolic functions in calculus

    Differentiating y=arcsin⁡xy=\arcsin x implicitly gives 11−x2\frac1{\sqrt{1-x^2}}; similar results hold for arctan⁡\arctan, arsinh⁡\operatorname{arsinh}, arcosh⁡\operatorname{arcosh}. Read backwards, these give the integrals of 1a2−x2\frac1{\sqrt{a^2-x^2}}, 1a2+x2\frac1{a^2+x^2} and 1x2±a2\frac1{\sqrt{x^2\pm a^2}}, which algebra alone can't do.

    ≈ 45 min

  2. Maclaurin series

    Any smooth function can be approximated near 00 by a polynomial whose derivatives match it there: f(x)=f(0)+f′(0)x+f′′(0)2!x2+⋯f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\cdots. Standard series for exe^x, sin⁡x\sin x, cos⁡x\cos x, ln⁡(1+x)\ln(1+x) and (1+x)n(1+x)^n combine by substitution, multiplication and differentiation.

    ≈ 45 min