Unit 7 - Objective 2 - Derivatives of the Other Trigonometric Functions



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

Derivative for tan(u)
Derivative for cot(u)
Derivative for sec(u)
Derivative for csc(u)


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





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