All the rules for derivatives (products, quotients, chain rule, and implicit differentiation) still apply for these trigonometric functions.

 


Problem 1: Find the derivative of y = 2tan 3x + sec x csc x
= 2 sec²3x (3x) + sec x (csc x) + csc x sec x
= (2 sec²3x) (3) + sec x (csc x cot x) + csc x (sec x tan x)
clean it up
= 6 sec²3x  (sec x csc x cot x) + (csc x sec x tan x)
Problem 2: Find the derivative of y = (cot 7x  sin 2x)³ (you need the power rule)
= 3 (cot 7x  sin 2x)² (cot 7x  sin 2x)
= 3 (cot 7x  sin 2x)² [csc² (7x) (7)  (cos 2x) (2)]
clean it up
= 3 (cot 7x  sin 2x)² (7 csc²7x  2 cos 2x)
Problem 3: Find the derivative of x tan y  4y = csc 4x
x (tan y) + tan y (x)  4 = csc 4x cot 4x (4x)
x sec²y + (tan y) (1)  4 = (csc 4x cot 4x) (4)
clean it up
x sec²y + tan y  4 =  4 (csc 4x cot 4x)
get by itself
x sec²y  4 =  4 (csc 4x cot 4x)  tan y
(x sec²y  4) =  4 (csc 4x cot 4x)  tan y