Derivative of trigonometric functions

Introduction to derivatives of trigonometric functions: Trigonometric functions are important because so many of the phenomena we want information about are periodic (electromagnetic fields, heart rhythms, tides, weather). A surprising and beautiful theorem from advanced calculus says that every periodic function we are likely to use in mathematical modelling can be written as an algebraic combination of sines and cosines, so the derivatives of sines and cosines play a key role in describing important changes.Let us understand derivative of six basic trigonometric functions .What are the Derivatives of Trig Functions?
(i) The derivative of the sine: The derivative of sine is cosine. d/dx (sin x) = cos x..
(ii) The derivative of cosine: The derivative of cosine is the negative of the sine. d/dx ( cos x) = - sin x.
Let us take an example to understand derivative of trig functions sine and cosine

Example 1: Find the derivative of y = x^2 – sin x.
Solution: dy/dx = 2x – d/dx (sin x)
= 2x – cos x.

Example 2: Find the derivative of y = 5x + cos x.
Solution: dy/dx = d/dx (5x) + d/dx ( cos x)
= 5 – sin x.
More Derivatives of Trig Functions: Because sin x and cos x are differentiable functions of x, the related functions:
tan x = sin x/cos x sec x = 1/cos x
cot x = cos x/sin x cosec x = 1/sin x
are differentiable at every value of x at which they are defined. Their derivatives calculated from the quotient rule are given by the formulas.
(iii) The  Derivative   of the tan  is :   d/dx (tan x) = sec^2 x    
(iv) The derivative of the sec is:   d/dx (sec x) = sec x . tan x
(v) The  Derivative   of the cot  is: d/dx ( cot x) = - cosec^2 x
(vi) The Derivative   of the cosec is  : d/dx ( cosec x) = - cosec x . cot x.
Let us understand more about Derivatives of Trig Functions through some examples.

Example 3: Find the derivative of 3sinx  + cot x.
Solution: here first we find Derivatives Trig Functions cotx and sinx  , we know d/dx cot x = -cosec^2x and we also know d/dx sinx = cosx. So we plug in problem:
d/dx (3sinx  + cot x) = 3 d/dx (sinx) + d/dx ( cot x) = 3cosx  – cosec^2 x.

Example 4: Find the derivative of 2/sin x.
Solution: d/dx ( 2 / sin x) = d/dx ( 2 / cosec x) = 2 d/dx (cosec x)
= 2 ( - cosec x . cot x ) = -2 . cosec x . cot x.

Example 5: Find the double derivative of y = sec x.
Solution: y = sec x
y ’ = sec x . tan x
y ” = d/dx ( sec x . tan x)
    = (sec x) d/dx (tan x) + (tan x) d/dx ( sec x)
    = sec x (sec^2 x) + tan x (sec x . tan x)
    = sec3 x + sec x . tan^2 x.