Derivatives of Trigonometric Functions

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]