Answers to Practice Problems on Differentiation Most of the answers are given in a MINIMALLY simplified form (such as combining the constants). If you solve the problem correctly, you should be able to see that your answer matches the answer posted. 1) 2x + 14(2x + 3)6 2) 25 (2x − 1)−4/5 3) cos(2x) − 2x sin(2x) 4) 2x cos(2x) − 2x2 sin(2x) √ − 3 x · sin(3x) 5) cos(3x) √ 2 x 6) 4√ 3 x cos(4x) 3 √ 3 − 4 x4 · sin(4x) 7) 2x · sin(4x) + 4x2 · cos(4x) 8) 8(2x − 3)3 · tan(4x) + 4(2x − 3)4 · sec2 (4x) 9) 4x3 sec(x2 + 1) + 2x5 sec(x2 + 1) tan(x2 + 1) 10) 2x cos(2x)−sin(2x) x2 11) cos 12) 1 2 2x+cos x x2 −7 sin 2x+1 x3 −6x · (2−sin x)(x2 −7)−2x(2x+cos x) (x2 −7)2 −1/2 · cos 2x+1 x3 −6x · 2(x3 −6x)−(2x+1)(3x2 −6) (x3 −6x)2 13) 15 sin2 (5x) · cos(5x) 14) 15 sec3 (5x) · tan(5x) 15) 15 sec3 (tan(5x)) · tan(tan(5x)) · sec2 (5x) 16) 4 tan3 17) 2 cos x sin x x sin x · sec2 x sin x · − (2x + 1) sin 18) (2+3 sec2 x) cos2 x sin x sin x−x cos x sin2 x x sin x · sin x−x cos x sin2 x −2(2x+3 tan x) cos x sin x ·sin x sin x · sin x−x cos x sin2 x 19) −24 cos2 (tan2 (4x)) · sin(tan2 (4x)) · tan(4x) · sec2 (4x) 20) √ 15 tan2 (5x−2)·sec2 (5x−2) x3 −sin x− 12 tan3 (5x−2)·(x3 −sin x)−1/2 ·(3x2 −cos x) x3 −sin x 21) 3 sec3 (sin(x2 +1)+2x)·tan(sin(x2 +1)+2x)·(2x cos(x2 +1)+2)· x3 +cos2 (3x) √ x3 +cos2 (3x)− − 12 (x3 +cos2 (3x))−1/2 ·(3x2 −6 cos(3x)·sin(3x)) sec3 (sin(x2 +1)+2x) x3 +cos2 (3x) 22) 3 sin2 √ · √ x x2 +cos x−6 cos(2x−1) 3x2 −sec(2x) · cos √ x x2 +cos x−6 cos(2x−1) 3x2 −sec(2x) · x2 +cos x−6 cos(2x−1)+ x2 (x2 +cos x−6 cos(2x−1))−1/2 ·(2x−sin x+12 sin(2x−1)) (3x2 −sec(2x))− (3x2 −sec(2x))2 √ −(6x−2 sec(2x) tan(2x))·x x2 +cos x−6 cos(2x−1) (3x2 −sec(2x))2 23) − cos(cos(tan(sec x))) · (sin(tan(sec x))) · (sec2 (sec x)) · sec x · tan x 24) − cos(cos x)·sin x·tan x·sec x−sin(cos x)·(sec3 x+tan2 x sec x) tan2 x·sec2 x