Extended Derivatives and Integrals (AA HL)
HL adds a longer list of standard functions on top of the SL toolkit: \(\tan x\), the reciprocal trig functions, \(a^x\), \(\log_a x\), and the inverse trig functions \(\arcsin x\), \(\arccos x\) and \(\arctan x\). Each has its own standard derivative, and the chain rule extends every one of them to composite arguments like \(\arctan(4x)\). This page covers the extra results, worked examples, and the mistakes that lose marks. It's part of the broader Integration topic.
11 questions on this sub-topic.
The extended standard results
Syllabus reference SL5.11 covers definite integrals, including the analytical approach \(\int_a^b g'(x)\,dx = g(b)-g(a)\), and areas enclosed by a curve and the \(x\)-axis. At HL those techniques get applied to a wider set of functions - the results below are what let you find the antiderivative in the first place, before evaluating it between limits.
Extended derivatives
\[\dfrac{d}{dx}(\tan x) = \sec^2 x,\quad \dfrac{d}{dx}(a^x) = a^x\ln a,\quad \dfrac{d}{dx}(\arctan x) = \dfrac{1}{1+x^2}\]
Also \(\arcsin x\) and \(\arccos x\) have standard-derivative forms involving \(\sqrt{1-x^2}\). Each reverses into a standard integral, so knowing the derivative unlocks the integral for free.
✓ In the formula bookletChain rule on composite arguments
\[\dfrac{d}{dx}\arctan(ax+b) = \dfrac{a}{1+(ax+b)^2}\]
Multiply the standard result by the derivative of the inner function, exactly as with \(\sin\), \(\cos\) and \(e^x\) at SL.
Not a listed formula - techniqueNeed the full syllabus wording and formula-booklet reference table? See Integration.
Worked examples
Find \(\dfrac{dy}{dx}\) for \(y=\arctan(4x).\)
Worked solution
\(\dfrac{d}{dx}\arctan(x) = \dfrac{1}{1+x^2}.\) M1
\(\dfrac{dy}{dx} = \dfrac{4}{1+(4x)^2}\) M1
\(= \dfrac{4}{1+16x^2}.\) A1
Find the following indefinite integrals.
(a) \(\displaystyle\int \sec^2 x\, dx\)
(b) \(\displaystyle\int \frac{1}{1+x^2}\, dx\)
(c) \(\displaystyle\int \frac{1}{\sqrt{1-x^2}}\, dx\)
(d) \(\displaystyle\int 2^x\, dx\)
Worked solution
(a) \(\tan x + C\) A1
(b) \(\arctan x + C\) A1
(c) \(\arcsin x + C\) A1
(d) \(\dfrac{2^x}{\ln 2} + C\) A1
Common mistakes
- Mixing up the arcsin and arctan derivatives. \(\arcsin x\) differentiates to something with a square root on the bottom, while \(\arctan x\) differentiates to \(\dfrac{1}{1+x^2}\) - confusing the two is one of the most common HL slips.
- Forgetting the chain-rule factor. When the argument isn't simply \(x\), such as \(\arctan(4x)\) or \(a^{2x+1}\), you must multiply by the derivative of the inner function - dropping this factor is an easy mark to lose under time pressure.
- Missing \(\ln a\) in the derivative of \(a^x\). \(\dfrac{d}{dx}(a^x) = a^x\ln a\), not just \(a^x\) - the \(\ln a\) factor is easy to forget when \(a\) isn't \(e\).
Ready to practise properly?
11 extended-derivatives-and-integrals questions, marked instantly like the real exam.
Quick answers
What extra derivatives does AA HL add beyond SL?
HL students also need the derivatives of \(\tan x\), \(\sec x\), \(\csc x\), \(\cot x\), \(a^x\), \(\log_a x\), \(\arcsin x\), \(\arccos x\) and \(\arctan x\), on top of the SL functions.
Are these extended derivatives in the formula booklet?
Yes - the AHL formula booklet lists the standard derivatives for \(\tan x\), \(a^x\), \(\log_a x\), \(\arcsin x\), \(\arccos x\) and \(\arctan x\), so you don't need to memorise them from scratch.