A LevelMaths: PureAQARevision Notes7. Differentiation7.3 Further Differentiation7.3.6 Differentiating Reciprocal and Inverse Trig Functions

Differentiating Reciprocal and Inverse Trig Functions (AQA A Level Maths: Pure)

Revision Note

Author

Roger

Last updated

30 July 2023

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Differentiating Reciprocal and Inverse Trig Functions

How do I differentiate the reciprocal trigonometric functions?

The formulae for the derivatives of the reciprocal trigonometric functions are:

\frac{d}{d x} (\sec x) = \sec x \tan x

\frac{d}{d x} (cosec x) = - cosec x \cot x

\frac{d}{d x} (\cot x) = - {cosec}^{2} x

You can derive the derivatives for sec, cosec, and cot using the chain rule and the derivatives of the basic trigonometric functions

Diff Rec Inv Trig Illustr 1, AS & A Level Maths revision notes

How do I differentiate the inverse trigonometric functions?

The formulae for the derivatives of the inverse trigonometric functions are:

\frac{d}{d x} (\arcsin x) = \frac{1}{\sqrt{1 - x^{2}}}

\frac{d}{d x} (arc \cos x) = - \frac{1}{\sqrt{1 - x^{2}}}

\frac{d}{d x} (arc \tan x) = \frac{1}{1 + x^{2}}

You can derive the derivatives of arcsin, arccos, and arctan using the reciprocal form of the chain rule and the derivatives of the basic trigonometric functions

Examiner Tip

The formulae for the derivatives of sec x, cosec x, and cot x are in the formulae booklet – you don't need to memorise them
- However, you should know how to derive the derivatives of sec, cosec, and cot using the chain rule
The formulae for the derivatives of arcsin, arccos, and arctan are not in the formulae booklet
- You should know how to derive those derivatives using the reciprocal form of the chain rule

Worked example

Diff Rec Inv Trig Example, AS & A Level Maths revision notes

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Author: Roger

Expertise: Maths

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.