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Derivatives of Trigonometric Functions Worksheet

Derivatives of Trigonometric Functions Worksheet
  • Page 1
 1.  
If y = (c(8c-sin 9c)3), then find dydc.
a.
(8c-sin 9c)2(8 - 9cos 9c)
b.
(3c) + 8c - sin 9c2c
c.
c(8-9cos 9c)
d.
(8c - sin 9c)2((8 - 9cos 9c)(3c) + 8c - sin 9c2c)
e.
3c(8c-sin 9c)2


Solution:

y = (c(8c-sin 9c)3)
[Write the function.]

dydc = ddc(c(8c - sin 9c)3)
[Find dydc.]

= cDc((8c - sin 9c)3)+(8c - sin 9c)3)Dc(c)
[Use the Product Rule.]

= c(3(8c - sin 9c)2)Dc(8c - sin 9c)+(8c - sin 9c)312c
[Use the Chain Rule.]

= 3c(8c - sin 9c)2(8 - 9cos 9c)+(8c - sin 9c)32c

= (8c - sin 9c)2((8 - 9cos 9c)(3c) + 8c - sin 9c2c)


Correct answer : (4)
 2.  
If y = 8b2sin (9b), then find dydb.
a.
(16b)sin(9b)
b.
- 72cos(9b)
c.
- (16b)sin(9b) + 72cos(9b)
d.
sin(9b) - cos(9b)
e.
(16b)sin(9b) - 72cos(9b)


Solution:

y = 8b2sin(9b)
[Write the function.]

dydb = ddb(8b2sin(9b))
[Find dydb.]

dydb = (8b2)Db(sin(9b))+sin (9b)Db(8b2)
[Use the Product Rule.]

= (8b2)(cos(9b))Db (9b)+sin(9b)(16b)
[Use the Chain Rule.]

= - (8b2)cos(9b)(9b2) + (16b)sin(9b)

= (16b)sin(9b) - 72cos(9b)

dydb = (16b)sin(9b) - 72cos(9b)


Correct answer : (5)
 3.  
If y = 12cosec5x, then find dydx.
a.
(3 ) (5x)(cot5x)(cosec5x)
b.
(5x)(cot5x)(cosec5x)
c.
- (5x)(cot5x)(cosec5x)
d.
(cot5x)(cosec5x)
e.
- (3 ) (5x)(cot5x)(cosec5x)


Solution:

y = 12cosec5x
[Write the function.]

dydx = ddx(12cosec5x)
[Find dydx.]

= 122cosec5xDx(cosec5x)
[Use the Chain Rule.]

= 122cosec5x(-(cosec5xcot5x)Dx(5x))
[Use the Chain Rule again.]

= - (6 ) cosec5x(cot5x)(52x)

= - (3 ) (5x)(cot5x)(cosec5x)

dydx = - (3 ) (5x)(cot5x)(cosec5x)


Correct answer : (5)
 4.  
If y = 5sin 5x7 + cos 5x, then find (7+cos 5x)2dydx.
a.
5(5cos 5x+cos 10x)
b.
25(7cos 5x+1)
c.
- 25(7cos 5x+1)
d.
25(7cos 5x+cos 10x)
e.
- 25(7cos 5x+cos 10x)


Solution:

y = 5sin 5x7 + cos 5x
[Write the function.]

dydx = d dx( 5 sin 5 x7+cos 5 x)
[Find dydx.]

= 5((7+cos 5x)ddx(sin 5x)-sin 5xddx(7+cos 5x)(7+cos 5 x)2)
[Use the Quotient Rule.]

= 5((7 + cos 5x)(5cos 5x) - sin 5x(-5sin 5x)(7+cos 5x)2)

= 5 (35cos5x+5(1)(7+cos 5x)2)
[Use sin² θ + cos² θ = 1.]

= 25(7cos5x+1(7+cos 5x)2)
[Simplify.]

dydx= 25(7cos5x+1(7+cos 5x)2)

(7+cos 5x)2dydx= 25(7cos5x+1)


Correct answer : (2)
 5.  
If g(m) = (5cos m + 6m)9, then find g′ (m).
a.
(54 - 45sin m)(5cos m + 6m)9
b.
(54 + 45sin m)(5cos m + 6m)8
c.
9(5cos m + 6m)8
d.
(6- 5 sin m)(5cos m + 6m)8
e.
(54 - 45sin m)(5cos m + 6m)8


Solution:

g (m) = (5cos m + 6m)9
[Write the function.]

g′ (m) = ddm ((5cos m + 6m)9)
[Find g′ (m).]

g′ (m) = 9(5cos m+6m)8ddm(5cos m+6m)
[Use the Chain Rule.]

= 9(5cos m+6m)8(- 5sin m+6)

= (54 - 45sin m)(5cos m + 6m)8
[Simplify.]

g′ (m) = (54 - 45sin m)(5cos m + 6m)8


Correct answer : (5)
 6.  
If g(n) = cos23n sin46n, then find g′ (n) .
a.
24(sin3 6n)(cos 6n)(cos23n)
b.
sin5 6n + (sin3 6n)(cos 6n)(cos23n)
c.
- 3(sin5 6n) + 24(sin3 6n)(cos 6n)(cos23n)
d.
3(sin5 6n)
e.
3(sin5 6n) - 24(sin3 6n)(cos 6n)(cos23n)


Solution:

g(n) = cos23n sin46n
[Write the function.]

g′ (n) = Dn(cos23n sin46n)
[Find g′ (n).]

= (sin46n)Dn(cos23n)+ (cos23n)Dn(sin46n)
[Use the Product Rule.]

= (sin46n)((2cos 3n)(-sin 3n)(3))+ 4sin 36n(cos 6n)(6)(cos23n)

= - 3(sin5 6n) + 24(sin3 6n)(cos 6n)(cos23n)
[Use 2sin θcos θ = sin 2θ and simplify.]

g′ (n) = - 3(sin5 6n) + 24(sin3 6n)(cos 6n)(cos23n)


Correct answer : (3)
 7.  
Find ′(u), if (u)= 2u2 + 4cot u3.
a.
4u + 12u2(cosec2 u3)
b.
12u2(cosec2 u3) - 4u
c.
4u - (cosec2 u3)
d.
4u - 12u2(cosec2 u3)
e.
12u2(cosec2 u3)


Solution:

(u)= 2u2 + 4cot u3
[Write the function.]

′(u) = ddu(2u2 + 4cot u3)
[Find f ′ (u).]

= 4u - 4(cosec2 u3)(3u2)
[Use the Chain Rule.]

= 4u - 12u2(cosec2 u3)
[Simplify.]

′(u) = 4u - 12u2(cosec2 u3)


Correct answer : (4)
 8.  
If g(k) = cosec 8k + 4, then find Dk [g(k)].
a.
cosec 8kcosec 8k + 4
b.
- 4(cosec 8k)(cot 8k)cosec 8k + 4
c.
(cosec 8k)(cot 8k)cosec 8k + 4
d.
- 4(cosec 8k)(cot 8k)cosec 8k - 4
e.
Does not exist


Solution:

g(k) = cosec 8k + 4
[Write the function.]

Dk (cosec 8k+4) = Dk((cosec 8k+4)12)
[Find Dk [g(k)] .]

= (-cosec 8kcot 8k)(8)2cosec 8k+4
[Use the Chain Rule.]

= -4(cosec 8k)(cot 8k)cosec 8k+4
[Simplify.]

Dk [g(k)] = -4(cosec 8k)(cot 8k)cosec 8k+4


Correct answer : (2)
 9.  
Find the derivative of the function
y = 1sin2 2x+cos2 4x

a.
III
b.
IV
c.
II
d.
I
e.
V


Solution:

y = 1sin2 2x+cos2 4x
[Write the function.]

dydx = ddx[1sin2 2x+cos2 4x]
[Find dydx.]

dydx = sin22x+cos24xDx(1)-Dxsin22x+cos24xsin22x+cos24x
[Use the Quotient Rule.]

= -12sin2 2x+cos2 4x 4sin 2x cos 2x-8cos 4x sin 4xsin2 2x+cos2 4x

= -12sin2 2x+cos2 4x(2sin 4x-4sin  8xsin2 2x+cos2 4x)
[Use 2sin θ cos θ = sin2 θ.]

= 2sin8x-sin4x(sin22x+cos24x)32

So, dydx = 2sin8x-sin4x(sin22x+cos24x)32


Correct answer : (4)
 10.  
If f(n) = sin2(3n2+2n)3, then find dfdn.
a.
(18n + 6)(cos 2(3n2 + 2n)3)
b.
(18n + 6)(3n2 + 2n)2(sin 2(3n2 + 2n)3)
c.
sin (3n2 + 2n)3
d.
(3n2 + 2n)2(sin 2(3n2 + 2n)3)
e.
(18n + 6)(3n2 + 2n)2


Solution:

f(n) = sin2(3n2 + 2n )3
[Write the function.]

dfdn =ddn(sin2(3n2 + 2n )3)
[Find dfdn.]

= 2sin (3n2 + 2n)3ddn(sin (3n2 + 2n)3)
[Use the Chain Rule.]

= 2sin (3n2 + 2n)3cos (3n2 + 2n)3(3)(3n2 + 2n)2(6n + 2)
[Use the Chain Rule again.]

= (18n + 6)(3n2 + 2n)2(sin 2(3n2 + 2n)3)
[Simplify.]

dfdn = (18n + 6)(3n2 + 2n)2(sin 2(3n2 + 2n)3)


Correct answer : (2)

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