Differentiating Reciprocal & Inverse Trig Functions (Edexcel IAL Maths)

Differentiating Reciprocal and Inverse Trig Functions

Spec point — 4.2

How do I differentiate reciprocal trigonometric functions?

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

$\frac{d}{dx}(\mathrm{sec}x)=\mathrm{sec}x\mathrm{tan}x$
$\frac{d}{dx}(\mathrm{cosec}x)=-\mathrm{cosec}x\mathrm{cot}x$ $\frac{d}{dx}(\mathrm{cot}x)=-\mathrm{cosec}^{2}x$

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

How do I differentiate inverse trigonometric functions?

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

$\frac{d}{dx}(\mathrm{arcsin}x)=\frac{1}{\sqrt{1-x^{2}}}$ $\frac{d}{dx}(\mathrm{arc} \mathrm{cos}x)=-\frac{1}{\sqrt{1-x^{2}}}$ $\frac{d}{dx}(\mathrm{arc} \mathrm{tan}x)=\frac{1}{1+x^{2}}$

• 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

Exam Hint

• 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