Trigonometric Differentiation Exercises-1
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Trigonometric Differentiation — Exercise Set I.
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Find the derivative f (x) for the following functions. Say what you can about the sign of f (x).
(I.1)
(I.2)
(I.3)
(I.4)
(I.5)
(I.6)
(I.7)
(I.8)
(I.9)
(I.10)
f (x) = tan3 (4x)
f (x) = tan (4x)3
f (x) = cot
√ x
f (x) = sec2 (2x + 1)2
f (x) = csc3 x − sec3 x
f (x) = x 3 tan(2x) − x 2 sec(3x)
f (x) = sec2 x csc x
1
f (x) = x tan
x
1
f (x) = x 2 tan
x
1
3
f (x) = x tan
x
Solution
Solution
Solution
Solution
Solution
Solution
Solution
Solution
Solution
Solution
Trigonometric Differentiation Exercises-2
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Trigonometric Differentiation — Exercise Set II.
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Find the derivative f (x) for the following functions. Say what you can about the sign of f (x).
(II.1)
(II.2)
(II.3)
(II.4)
(II.5)
(II.6)
(II.7)
(II.8)
(II.9)
(II.10)
f (x) = x tan x
Solution
f (x) = tan2 x
Solution
f (x) = tan3 x
Solution
f (x) = tan4 x
Solution
f (x) = tan5 x
Solution
f (x) =
1 − cot x
1 + cot x
Solution
f (x) =
1 + cot x
1 − cot x
Solution
f (x) =
1 − tan x
1 + tan x
Solution
f (x) =
1 + tan x
1 − tan x
Solution
f (x) =
tan x
1 − tan x
Solution
Trigonometric Differentiation Exercises-3
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Trigonometric Differentiation — Exercise Set III.
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Find the derivative f (x) for the following functions. Say what you can about the sign of f (x).
(III.1)
(III.2)
(III.3)
(III.4)
(III.5)
(III.6)
(III.7)
(III.8)
(III.9)
(III.10)
f (x) = sin(sin x)
Solution
f (x) = tan2 (cos x)
Solution
f (x) = cot(tan3 x)
Solution
f (x) = csc(cot4 x)
Solution
f (x) = sec(csc5 x)
Solution
f (x) = sin(sin(sin x))
f (x) = cos(cos x)
Solution
Solution
f (x) = cos(cos(cos x))
Solution
f (x) = tan(cot(tan x))
Solution
1
f (x) = csc
1 + x2
Solution
Trigonometric Differentiation Exercises-4
f (x) = tan3 (4x)
Solution:
(I.2)
f (x) = 3 tan2 (4x)(tan 4x) = 3 tan2 (4x) sec2 (4x)(4x) = 12 tan2 (4x) sec2 (4x) ≥ 0
f (x) = tan (4x)3
f (x) = sec2 (4x)3 (4x)3 = sec2 (4x)3 3(4x)2 (4x) = sec2 (4x)3 3(4x)2 (4) =
12(4x)2 sec2 (4x)3 ≥ 0
Solution:
(I.3)
f (x) = cot
f (x) = − csc
2
√ x
1
f (x) = cot x 2
Solution:
x
1
2
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(I.1)
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Solution Set I
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x
1
2
2
= − csc
x
1
2
1
√ 1
1
x − 2 = − √ csc2
x <0
2
2 x
Trigonometric Differentiation Exercises-5
(I.4)
f (x) = sec2 (2x + 1)2
f (x) = 2 sec (2x + 1)2 sec (2x + 1)2
=
2 sec (2x + 1)2 sec (2x + 1)2 tan (2x + 1)2 (2x + 1)2 =
2 sec2 (2x + 1)2 tan (2x + 1)2 2(2x + 1)(2x + 1) =
2 sec2 (2x + 1)2 tan (2x + 1)2 2(2x + 1)(2) = 8(2x + 1) sec2 (2x + 1)2 tan (2x + 1)2
Solution:
(I.5)
f (x) = csc3 x − sec3 x
Solution:
f (x) = 3 csc2 x(csc x) − 3 sec2 x(sec x) =
3 csc2 x(− csc x cot x) − 3 sec2 x(sec x tan x) = −3 csc3 x cot x − 3 sec3 x tan x
(I.6)
f (x) = x 3 tan(2x) − x 2 sec(3x)
Solution:
f (x) = (x 3 ) tan(2x) + x 3 (tan(2x)) − (x 2 ) sec(3x) − x 2 (sec(3x)) =
(3x 2 ) tan(2x) + x 3 (sec2 (2x))(2x) − (2x) sec(3x) − x 2 (sec(3x) tan(3x))(3x) =
(3x 2 ) tan(2x) + x 3 (sec2 (2x))(2) − (2x) sec(3x) − x 2 (sec(3x) tan(3x))(3) =
(3x 2 ) tan(2x) + 2x 3 (sec2 (2x)) − 2x sec(3x) − 3x 2 (sec(3x) tan(3x))
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Trigonometric Differentiation Exercises-6
(I.7)
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f (x) = sec2 x csc x
f (x) = sec2 x csc x + sec2 x (csc x) = 2 sec x (sec x) csc x + sec2 x(− csc x cot x) =
sin x 1
−
2 sec x(sec x tan x) csc x − sec2 x csc x cot x = 2 sec2 x tan x csc x − sec2 x csc x cot x = 2 sec2 x
cos x sin x
1 cos x
1
= 2 sec2 x
− sec x csc2 x = 2 sec3 x − sec x csc2 x = sec x 2 sec2 x − csc2 x
sec2 x
sin x sin x
cos x
1
f (x) = x tan
x
Solution:
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1
1
1
1
2 1
f (x) = (x) tan
+ x tan
+ x sec
= tan
=
x
x
x
x
x
1
1
1
1 −1
1
tan
+ x sec2
− sec2
= tan
2
x
x x
x
x
x
(I.9)
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Solution:
(I.8)
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f (x) = x 2 tan
1
x
1
1
1
1
2
2
2 1
+ x tan
+ x sec
Solution: f (x) = (x ) tan
= 2x tan
=
x
x
x
x
x
1
−1
1
2
2 1
2 1
2x tan
+ x sec
− sec
= 2x tan
x
x x2
x
x
2 Trigonometric Differentiation Exercises-7
(I.10)
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1
f (x) = x tan
x
1
1
1
1
3
2
3
2 1
+ x tan
+ x sec
Solution: f (x) = (x ) tan
= 3x tan
=
x
x
x
x
x
−1
1
1
2
3
2 1
2
2 1
+ x sec
− x sec
= 3x tan
3x tan
x
x x2
x
x
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Trigonometric Differentiation Exercises-8
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(II.1)
f (x) = x tan x
Solution:
(II.2)
f (x) = tan2 x
Solution:
(II.3)
f (x) = (x) tan x + x(tan x) = tan x + x sec2 x
f (x) = 2 tan x(tan x) = 2 tan x sec2 x
f (x) = tan3 x
Solution:
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f (x) = 3 tan2 x(tan x) = 3 tan2 x sec2 x > 0
(II.4)
f (x) = tan4 x
Solution:
f (x) = 4 tan3 x(tan x) = 4 tan3 x sec2 x
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Solution Set II
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Trigonometric Differentiation Exercises-9
(II.5)
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f (x) = tan5 x
f (x) = 5 tan4 x(tan x) = 5 tan4 x sec2 x > 0
f (x) =
Solution:
1 − cot x
1 + cot x
f (x) =
(1 − cot x) (1 + cot x) − (1 − cot x)(1 + cot x)
=
(1 + cot x)2
−(− csc2 x)(1 + cot x) − (1 − cot x)(− csc2 x)
(1 + cot x) + (1 − cot x)
2 csc2 x
2
=
csc
x
=
>0
(1 + cot x)2
(1 + cot x)2
(1 + cot x)2
(II.7)
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Solution:
(II.6)
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f (x) =
1 + cot x
1 − cot x
Solution:
f (x) =
(1 + cot x) (1 − cot x) − (1 + cot x)(1 − cot x)
− csc2 x(1 − cot x) − (1 + cot x)(−(− csc2 x))
=
=
(1 − cot x)2
(1 − cot x)2
− csc2 x
(1 − cot x) + (1 + cot x)
2 csc2 x
=
−
<0
(1 − cot x)2
(1 − cot x)2
Trigonometric Differentiation Exercises-10
Solution:
f (x) =
sec2 x
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(1 − tan x) (1 + tan x) − (1 − tan x)(1 + tan x)
− sec2 x(1 + tan x) − (1 − tan x)(sec2 x)
=
=
(1 + tan x)2
(1 + tan x)2
−(1 + tan x) − (1 − tan x)
−2 sec2 x
=
(1 + tan x)2
(1 + tan x)2
(II.9)
f (x) =
f (x) =
1 + tan x
1 − tan x
(1 + tan x) (1 − tan x) − (1 + tan x)(1 − tan x)
sec2 x(1 − tan x) − (1 + tan x)(− sec2 x)
=
=
(1 − tan x)2
(1 − tan x)2
2 sec2 x
(1 − tan x)2
(II.10)
f (x) =
tan x
1 − tan x
Solution:
f (x) =
sec2 x
(tan x) (1 − tan x) − tan x(1 − tan x)
sec2 x(1 − tan x) − tan x(− sec2 x)
=
=
>0
(1 − tan x)2
(1 − tan x)2
(1 − tan x)2
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(II.8)
1 − tan x
f (x) =
1 + tan x
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Trigonometric Differentiation Exercises-11
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(III.1)
Solution:
(III.2)
Solution:
f (x) = sin(sin x)
f (x) = (cos(sin x))(sin x) = (cos(sin x))(cos x) = cos x cos(sin x)
f (x) = tan2 (cos x)
f (x) = 2 tan(cos x)(tan(cos x)) = 2 tan(cos x)(sec2 (cos x))(cos x) =
2 tan(cos x)(sec2 (cos x))(− sin x) = −2 sin x tan(cos x) sec2 (cos x)
(III.3)
Solution:
f (x) = cot(tan3 x)
f (x) = − csc2 (tan3 x)(tan3 x) = − csc2 (tan3 x)(3 tan2 x(tan x) ) =
− csc2 (tan3 x)(3 tan2 x(sec2 x) = −3 tan2 x sec2 x csc2 (tan3 x) < 0
(III.4)
Solution:
f (x) = csc(cot4 x)
f (x) = − csc(cot4 x) cot(cot4 x)(cot4 x) = − csc(cot4 x) cot(cot4 x)(4 cot3 x)(cot x) =
− csc(cot4 x) cot(cot4 x)(4 cot3 x)(− csc2 x) = 4 cot3 x csc2 x csc(cot4 x) cot(cot4 x)
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Solution Set III
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Trigonometric Differentiation Exercises-12
(III.5)
Solution:
Solution:
f (x) = sec(csc5 x)
f (x) = sec(csc5 x) tan(csc5 x)(csc5 x) = sec(csc5 x) tan(csc5 x)(5 csc4 x)(csc x) =
f (x) = sin(sin(sin x))
f (x) = cos(sin(sin x))(sin(sin x)) = cos(sin(sin x))(cos(sin x))(sin x) =
cos(sin(sin x))(cos(sin x))(cos x) = cos x cos(sin x) cos(sin(sin x))
(III.7)
Solution:
(III.8)
Solution:
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sec(csc5 x) tan(csc5 x)(5 csc4 x)(− csc x cot x) = −5 csc5 x cot x sec(csc5 x) tan(csc5 x)
(III.6)
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f (x) = cos(cos x)
f (x) = − sin(cos x)(cos x) = − sin(cos x)(− sin x) = sin x sin(cos x)
f (x) = cos(cos(cos x))
f (x) = − sin(cos(cos x))(cos(cos x)) = − sin(cos(cos x))(− sin(cos x))(cos x) =
− sin(cos(cos x))(− sin(cos x))(− sin x) = − sin x sin(cos x) sin(cos(cos x))
Trigonometric Differentiation Exercises-13
(III.9)
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f (x) = tan(cot(tan x))
f (x) = sec2 (cot(tan x))(cot(tan x)) = sec2 (cot(tan x))(− csc2 (tan x))(tan x) =
sec2 (cot(tan x))(− csc2 (tan x))(sec2 x) = − sec2 x csc2 (tan x) sec2 (cot(tan x)) < 0
(III.10)
Solution:
1
f (x) = csc
1 + x2
1
1
1
1
2 −1
2 −2
f (x) = − csc
)
=
−
csc
)
(2x)
=
cot
(1
+
x
cot
(−1)(1
+
x
1 + x2
1 + x2
1 + x2
1 + x2
1
1
−4x
csc
cot
(1 + x 2 )2
1 + x2
1 + x2
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