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