CEGEP CHAMPLAIN - ST. LAWRENCE 201-NYA-05: Differential Calculus Patrice Camiré Derivatives of Trigonometric Functions 1. Find the derivative of the following functions. (a) sin(3x) (d) sec(−2x) (g) 2 sin(x3 ) (j) sec(x9 ) (b) cos(−4x) (e) cot(6x) (k) cot(x3 + x − 2) (c) tan(8x) (f) csc(−9x) (h) 8 cos(1 − x2 ) √ (i) tan( x) (l) csc(x10 ) 2. Find the derivative of the following functions. (a) sin(cos(2x)) (c) tan(cos(x2 )) (b) cos(4 + sin(3x)) (d) sec(tan(x)) (e) cot(sin(π 2 )) (f) csc(3 cos(2x)) 3. Find the derivative of the following functions. (a) x2 sin(x) (c) x3 cos(6x) (b) sin(x) cos(x) (d) cos(x) tan(x) sin(x) x p (f) 3 cos(9x) (e) 4. Find the derivative of the following functions. x+1 (a) sin2 (x3 + 1) (c) tan x−1 (b) cos3 (2x + 7) (d) sin(x cos(x)) p tan(x)) 4 2x − 1 (f) sin x+3 (e) cos( 5. Find the derivative of the following functions. (a) sin(x) 1 + cos(x) (c) tan(x) 1 + sin(x) (b) x sin(x) 1 + cos(x) (d) 1 + sin(x) 1 + cos(x) sin2 (3x) x4 tan(x) (f) 1 − sec(x) (e) 6. Find the derivative of the following functions. p sin(x2 ) sec(3x) (a) x sin(x) cos(x) (c) (b) x2 sin(2x) tan(x) (d) x4 sec(x) tan(x) Answers 1. (a) 3 cos(3x) (b) 4 sin(−4x) (c) 8 sec2 (8x) (d) −2 sec(−2x) tan(−2x) (e) −6 csc2 (6x) (f) 9 csc(−9x) cot(−9x) (g) 6x2 cos(x3 ) (j) 9x8 sec(x9 ) tan(x9 ) (k) −(3x2 + 1) csc2 (x3 + x − 2) 2 (h) 16x sin(1 − x ) √ sec2 ( x) √ (i) 2 x (l) −10x9 csc(x10 ) cot(x10 ) (d) sec(tan(x)) tan(tan(x)) sec2 (x) 2. (a) −2 sin(2x) cos(cos(2x)) (b) −3 cos(3x) sin(4 + sin(3x)) (e) 0 (c) −2x sin(x2 ) sec2 (cos(x2 )) (f) 6 sin(2x) csc(3 cos(2x)) cot(3 cos(2x)) 3. (a) x(2 sin(x) + x cos(x)) (b) 1 − 2 sin2 (x) 2 (c) 3x (cos(6x) − 2x sin(6x)) 4. (a) 6x2 sin(x3 + 1) cos(x3 + 1) (b) −6 sin(2x + 7) cos2 (2x + 7) −2 2 x+1 sec (c) (x − 1)2 x−1 (d) cos(x) (e) (f) x cos(x) − sin(x) x2 −3 sin(9x) [cos(9x)]2/3 (d) cos(x cos(x))[cos(x) − x sin(x)] p − sec2 (x) sin( tan(x)) p (e) 2 tan(x) 28 2x − 1 3 2x − 1 (f) sin cos (x + 3)2 x+3 x+3 5. (a) 1 1 + cos(x) (d) 1 + sin(x) + cos(x) (1 + cos(x))2 (b) x + sin(x) 1 + cos(x) (e) 2 sin(3x)[3x cos(3x) − 2 sin(3x)] x5 (c) 1 + sin3 (x) [(1 + sin(x)) cos(x)]2 (f) sec(x) sec(x) − 1 6. (a) sin(x) cos(x) + x cos2 (x) − x sin2 (x) = sin(x) cos(x) + x(2 cos2 (x) − 1) (b) x[2 sin(2x) tan(x) + 2x cos(2x) tan(x) + x sin(2x) sec2 (x)] p sec(3x)[2x cos(x2 ) + 3 sin(x2 ) tan(3x)] p (c) 2 sin(x2 ) (d) x3 sec(x)[4 tan(x) + x tan2 (x) + x sec2 (x)] = x3 sec(x)[4 tan(x) + 2x tan2 (x) + x]