AQA A Level Maths: Pure复习笔记7.3.1 First Principles Differentiation - Trigonometry

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

7.3.1-1st-Princ-Trig-Diff-Illustr-3_forms

7.3.1-1st-Princ-Trig-Diff-Illustr-3_derivs

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

7.3.1-1st-Princ-Trig-Diff-Illustr-4_forms

7.3.1-1st-Princ-Trig-Diff-Illustr-4_derivs

Derivatives of other trigonometric functions

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

7.3.1-1st-Princ-Trig-Diff-Illustr-5

  • 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:7.3.1-1st-Princ-Trig-Diff-Illustr-6
  • These formulae follow from combining the derivatives of the three basic functions with the chain rule, but they are worth knowing on their own

Exam Tip

  • Remember that when doing calculus with trigonometric functions you have to measure angles in radians.
  • The formula for the derivative of tan x is included in the exam formulae booklet.
  • The derivatives of sin x and cos x are NOT included in the formula booklet – you have to know them.
  • The small angle approximations for cos xsin x and tan x are included in the exam formulae booklet – you don't have to memorise them.
  • Be sure to read first principle differentiation exam questions clearly – they will state any results you can treat as 'givens' in your answer.

Worked Example

7.3.1-1st-Princ-Trig-Diff-Example

 

 

 

 

 

 

转载自savemyexams

更多Alevel课程
翰林国际教育资讯二维码