7.3.1 First Principles Differentiation - Trigonometry

How do I derive the 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:

• 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

• 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

 

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