Infinite Calculus - Limits and Derivatives of Trig Functions
[Pages:8]Calculus
Name___________________________________
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Limits and Derivatives of Trig Functions
Evaluate each limit.
1) lim -sec (x)
3p x?-
4
2) lim cos (x)
x ? -p
3) lim csc (2x)
5p x?
6
4) lim -2sin (2x)
3p x?
4
5) lim -2sin (x)
p x?-
2
7) lim -tan (x)
p x?
3
9) lim cot (x)
3p x?
4
Differentiate each function with respect to x.
11) y = sin x4
6) lim 2cos (x)
p x?-
6
8) lim sec (2x)
x ? -p
10) lim -sin (2x)
5p x?
6
12) y = cos 4x5
13) y = cos 3x3
14) y = sin 3x2
15) y = sec 2x4
16) y = tan x5
17) y = cot 2x4
18) y = csc x2
19) y = (2x4 - 5)sin 3x3
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20) y = cos 3x5 ? (5x2 - 1) 21) y = (-x5 + 5)sec x2 22) y = tan 3x2 ? (4x3 + 5) 23) y = (-5x4 - 3)csc 5x3 24) y = cot 2x2 ? (-x5 + 5) 25) y = sec 3x3 ? (2x5 - 5)
-5x5 - 1 26) y = cos 5x2
x3 - 5 27) y = tan x2
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5x2 28) y = sin -3x5 + 1
29) y = sin (sin 2x2) 31) y = cos (sin 2x4) 33) y = cot (sec 3x4) 34) y = cot (cot 4x2) 36) y = cot (cot x2)
30) y = cos (cos 3x4) 32) y = sin (cos 5x4)
35) y = tan (cot 2x5)
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37) y = sec (sec 5x5) 38) y = sin (2x2 - 5)3 40) y = sec (2x2 + 1)3
41) y = cot x3
39) y = cos (-3x3 + 4)2
42) y = cos2 3x2
43) y = sec3 2x2
44) y = sin3 3x4
45) y = cot2 2x3
46) y = tan (-5x4 - 4)2
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Worksheet by Kuta Software LLC
Calculus
Name___________________________________
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Limits and Derivatives of Trig Functions
Evaluate each limit.
1) lim -sec (x) 2
3p x?-
4
3)
lim csc (2x)
2 -
3
5p x?
3
6
2) lim cos (x) -1
x ? -p
4) lim -2sin (2x) 2
3p x?
4
5) lim -2sin (x) 2
p x?-
2
7) lim -tan (x) - 3
p x?
3
9) lim cot (x) -1
3p x?
4
Differentiate each function with respect to x.
11) y = sin x4
dy = cos x4 ? 4x3 dx
= 4x3cos x4
6) lim 2cos (x) 3
p x?-
6
8) lim sec (2x) 1
x ? -p
10) lim -sin (2x) 3
5p x?
2
6
12) y = cos 4x5
dy = -sin 4x5 ? 20x4 dx
= -20x4sin 4x5
13) y = cos 3x3
dy = -sin 3x3 ? 9x2 dx
= -9x2sin 3x3
14) y = sin 3x2
dy = cos 3x2 ? 6x dx
= 6xcos 3x2
15) y = sec 2x4
dy = sec 2x4tan 2x4 ? 8x3 dx
= 8x3sec 2x4tan 2x4
16) y = tan x5
dy = sec2 x5 ? 5x4 dx
= 5x4sec2 x5
17) y = cot 2x4
dy = -csc2 2x4 ? 8x3 dx
= -8x3csc2 2x4
18) y = csc x2
dy = -csc x2cot x2 ? 2x dx
= -2xcsc x2cot x2
19) y = (2x4 - 5)sin 3x3
dy = (2x4 - 5) ? cos 3x3 ? 9x2 + sin 3x3 ? 8x3
dx
= x2(18x4cos 3x3 - 45cos 3x3 + 8xsin 3x3)
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20) y = cos 3x5 ? (5x2 - 1)
dy = cos 3x5 ? 10x + (5x2 - 1) ? -sin 3x5 ? 15x4
dx
= 5x(2cos 3x5 - 15x5sin 3x5 + 3x3sin 3x5)
21) y = (-x5 + 5)sec x2
dy = (-x5 + 5) ? sec x2tan x2 ? 2x + sec x2 ? -5x4
dx
= xsec x2 ? (-2x5tan x2 + 10tan x2 - 5x3)
22) y = tan 3x2 ? (4x3 + 5) dy = tan 3x2 ? 12x2 + (4x3 + 5) ? sec2 3x2 ? 6x
dx
= 6x(2xtan 3x2 + 4x3sec2 3x2 + 5sec2 3x2)
23) y = (-5x4 - 3)csc 5x3
dy = (-5x4 - 3) ? -csc 5x3cot 5x3 ? 15x2 + csc 5x3 ? -20x3
dx
= 5x2csc 5x3 ? (15x4cot 5x3 + 9cot 5x3 - 4x)
24) y = cot 2x2 ? (-x5 + 5) dy = cot 2x2 ? -5x4 + (-x5 + 5) ? -csc2 2x2 ? 4x
dx
= x(-5x3cot 2x2 + 4x5csc2 2x2 - 20csc2 2x2)
25) y = sec 3x3 ? (2x5 - 5)
dy = sec 3x3 ? 10x4 + (2x5 - 5) ? sec 3x3tan 3x3 ? 9x2
dx
= x2sec 3x3 ? (10x2 + 18x5tan 3x3 - 45tan 3x3)
-5x5 - 1 26) y = cos 5x2
dy cos 5x2 ? -25x4 - (-5x5 - 1) ? -sin 5x2 ? 10x
=
dx
cos2 5x2
5x(-5x3cos 5x2 - 10x5sin 5x2 - 2sin 5x2)
= cos2 5x2
x3 - 5 27) y = tan x2
dy tan x2 ? 3x2 - (x3 - 5) ? sec2 x2 ? 2x
=
dx
tan2 x2
x(3xtan x2 - 2x3sec2 x2 + 10sec2 x2)
= tan2 x2
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5x2 28) y = sin -3x5 + 1
dy
5x2 (-3x5 + 1) ? 10x - 5x2 ? -15x4
dx = cos -3x5 + 1 ?
(-3x5 + 1)2
5xcos
5x2 -3x5 +
1
?
(9x5
+
2)
=
(-3x5 + 1)2
29) y = sin (sin 2x2)
30) y = cos (cos 3x4)
dy = cos (sin 2x2) ? cos 2x2 ? 4x
dx
= 4xcos (sin 2x2)cos 2x2
dy = -sin (cos 3x4) ? -sin 3x4 ? 12x3
dx
= 12x3sin (cos 3x4)sin 3x4
31) y = cos (sin 2x4)
dy = -sin (sin 2x4) ? cos 2x4 ? 8x3
dx
= -8x3sin (sin 2x4)cos 2x4
32) y = sin (cos 5x4)
dy = cos (cos 5x4) ? -sin 5x4 ? 20x3
dx
= -20x3cos (cos 5x4)sin 5x4
33) y = cot (sec 3x4)
dy = -csc2 (sec 3x4) ? sec 3x4tan 3x4 ? 12x3
dx
= -12x3csc2 (sec 3x4)sec 3x4tan 3x4
34) y = cot (cot 4x2)
dy = -csc2 (cot 4x2) ? -csc2 4x2 ? 8x
dx
= 8xcsc2 (cot 4x2)csc2 4x2
35) y = tan (cot 2x5)
dy = sec2 (cot 2x5) ? -csc2 2x5 ? 10x4
dx
= -10x4sec2 (cot 2x5)csc2 2x5
36) y = cot (cot x2)
dy = -csc2 (cot x2) ? -csc2 x2 ? 2x
dx
= 2xcsc2 (cot x2)csc2 x2
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37) y = sec (sec 5x5)
dy = sec (sec 5x5)tan (sec 5x5) ? sec 5x5tan 5x5 ? 25x4
dx
= 25x4sec (sec 5x5)tan (sec 5x5)sec 5x5tan 5x5
38) y = sin (2x2 - 5)3
dy = cos (2x2 - 5)3 ? 3(2x2 - 5)2 ? 4x
dx
= 12xcos (2x2 - 5)3 ? (2x2 - 5)2
39) y = cos (-3x3 + 4)2
dy = -sin (-3x3 + 4)2 ? 2(-3x3 + 4) ? -9x2
dx
= 18x2sin (-3x3 + 4)2 ? (-3x3 + 4)
40) y = sec (2x2 + 1)3
dy = sec (2x2 + 1)3tan (2x2 + 1)3 ? 3(2x2 + 1)2 ? 4x
dx
= 12xsec (2x2 + 1)3tan (2x2 + 1)3 ? (2x2 + 1)2
41) y = cot x3
1
dy
=
1
?
(cot
-
x3)
2
?
-csc2
x3
?
3x2
dx 2
3x2csc2 x3 = -
1
2(cot x3) 2
43) y = sec3 2x2
dy = 3sec2 2x2 ? sec 2x2tan 2x2 ? 4x dx
= 12xsec3 2x2tan 2x2
42) y = cos2 3x2
dy = 2cos 3x2 ? -sin 3x2 ? 6x dx
= -12xcos 3x2sin 3x2
44) y = sin3 3x4
dy = 3sin2 3x4 ? cos 3x4 ? 12x3 dx
= 36x3sin2 3x4cos 3x4
45) y = cot2 2x3
dy = 2cot 2x3 ? -csc2 2x3 ? 6x2 dx
= -12x2cot 2x3csc2 2x3
46) y = tan (-5x4 - 4)2
dy = sec2 (-5x4 - 4)2 ? 2(-5x4 - 4) ? -20x3
dx
= -40x3sec2 (-5x4 - 4)2(-5x4 - 4)
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