Differentiation of Trigonometric Functions from First Principles (Edexcel A Level Maths: Pure): Revision Note

Differentiation of trigonometric functions from first principles

How do I find derivatives of trigonometric functions from first principles?

• Recall that for a function f(x) the definition of the derivative from first principles (see First Principles Differentiation) is:

$\mathrm{f}\text{'}\left(x\right)=\underset{\mathrm{h}\to 0}{\lim }\frac{\mathrm{f}(x+h)-\mathrm{f}\left(x\right)}{h}$

• The derivatives of the trigonometric functions depend on the following small angle approximations

  • When θ is  small (i.e. close to zero) and measured in radians then

$$\begin{gathered} \sin \theta \approx \theta \\ \cos \theta \approx 1-\frac{1}{2}{\theta }^2 \\ \tan \theta \approx \theta \end{gathered}$$

• The small angle approximations allow us to produce the following intermediate limit results:

• And those intermediate results allow us to find the derivatives of sin and cos

How do I find derivatives of other trigonometric functions?

• The derivative of tan is given by the following formula:

• The easiest way to derive this is to use the quotient rule and the derivatives of sin and cos

• But it can also be derived from first principles using the small angle approximation for tan (see the Worked Example)

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

  

• These formulae follow from combining the derivatives of the three basic functions with the chain rule, but they are worth knowing on their own