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Derivatives of Hyperbolic Functions Worksheet

Derivatives of Hyperbolic Functions Worksheet
  • Page 1
 1.  
Find the derivative of f(x) = 7x5 cosh 6x.
a.
7x4 ( - x sinh 6x + 5cosh 6x)
b.
7x5 (6sinh 6x + x cosh 6x)
c.
7x4 (6x sinh 6x + 5cosh 6x)
d.
7x4 (6x sinh 6x + cosh 6x)
e.
35x4 sinh 6x


Solution:

f(x) = 7x5 cosh 6x
[Write the function.]

f ′(x) = ddx (7x5 cosh 6x)
[Find f ′(x).]

= 7x5 ddx (cosh 6x) + cosh 6x ddx (7x5)
[Use the Product Rule.]

= 42x5 sinh 6x + 35x4 cosh 6x

= 7x4 (6x sinh 6x + 5cosh 6x)
[Factor.]

The derivative of f(x) = 7x5 cosh 6x is 7x4 (6x sinh 6x + 5cosh 6x)


Correct answer : (3)
 2.  
Find the derivative of g(x) = e8x+x2 sinh 7x.
a.
8e8x + x (7x cosh 7x + 2sinh 7x)
b.
e8x + 14x cosh 7x
c.
e8x x cosh 7x
d.
e8x + x (- 7x cosh 7x + 2sinh 7x)
e.
e8x + x (7x cosh 7x + sinh 7x)


Solution:

g(x) = e8x+x2 sinh 7x
[Write the function.]

g ′(x) = ddx (e8x+x2 sinh 7x)
[Find f ′(x).]

g ′(x) = 8e8x + x2 (7cosh 7x) + sinh 7x(2x)
[Use Sum Rule, Product Rule.]

g ′(x) = 8e8x + x(7x cosh 7x + 2sinh 7x)
[Factor.]

The derivative of g(x) = e8x+x2 sinh 7x is 8e8x + x(7x cosh 7x + 2sinh 7x)


Correct answer : (1)
 3.  
Which of the following is the derivative of g(x) = 3tanh 8x + sech 8x?
a.
sech 8x (3tanh 8x - sech 8x)
b.
8sech x (3sech 8x - tanh 8x)
c.
sech 8x (tanh 8x - 3sech 8x)
d.
sech 8x (3tanh 8x + sech 8x)
e.
sech 8x (3sech 8x + tanh 8x)


Solution:

g(x) = 3tanh 8x + sech 8x
[Write the function.]

g ′(x) = ddx (3tanh 8x + sech 8x)
[Find g ′(x).]

g ′(x) = ddx (3tanh 8x) + ddx (sech 8x)
[Use Sum Rule.]

g ′(x) = 24sech2 8x + (- 8sech 8x tanh 8x)

g ′(x) = 8sech 8x (3sech 8x - tanh 8x)
[Factor.]

The derivative of g(x) = 3tanh 8x + sech 8x is 8sech 8x (3sech 8x - tanh 8x)


Correct answer : (2)
 4.  
Which of the following is the derivative of (x) = ecosh2 2x?
a.
- ecosh2 2x (sinh 4x)
b.
2ecosh2 2x (sinh 4x)
c.
4ecosh2 2x (cosh 2x)
d.
ecosh2 4x
e.
4ecosh2 2x (cosh 4x)


Solution:

(x) = ecosh2 2x
[Write the function.]

′(x) = ddx (ecosh2 2x)
[Find ′(x).]

= ecosh2 2x (2cosh 2x) (sinh 2x)(2)
[Use Chain Rule.]

= 2ecosh2 2x (sinh 4x)
[Use 2sinh x cosh x = sinh 2x]

The derivative of (x) = ecosh2 2x is 2ecosh2 2x (sinh 4x)


Correct answer : (2)
 5.  
What is the derivative of y = sinh 9tt+6?
a.
9cosh 9tt+6-sinh 9t(t+6)2
b.
cosh t - sinh t(t+1)2
c.
cosh 9t
d.
- cosh 9tt+6-sinh 9tt+6
e.
cosh 9tt+6+sinh 9t(t+6)2


Solution:

y = sinh 9tt+6
[Write the function.]

dydt = ddt (sinh 9tt+6)
[Find dydt.]

dydt = (t+6)(9cosh 9t) -sinh 9t (1)(t+6)2
[Use Quotient Rule.]

= 9cosh 9tt+6-sinh 9t(t+6)2

The derivative of y = sinh 9tt+6 is 9cosh 9tt+6-sinh 9t(t+6)2


Correct answer : (1)
 6.  
Find dydx, if y = 1 4 cosh x - x29.
a.
1 4 sinh x - x
b.
- 1 4 sinh x - x
c.
1 4 sinh x - 2 9x
d.
1 4 sinh x - 2 9x3
e.
1 4 sinh x - x3


Solution:

y = 1 / 4 cosh x - x29
[Write the function.]

dydx = ddx(1 / 4 cosh x - x29)
[Find dydx.]

= 1 / 4 sinh x - 2x9
[Use Sum Rule.]

= 1 / 4 sinh x - 2 / 9x
[Simplify.]

Therefore, dydx = 1 / 4 sinh x - 2 / 9x


Correct answer : (3)
 7.  
Find the derivative of (x) = tanh- 1 (sin 4x).
a.
4cosec2 4x
b.
4sec 4x
c.
- sec 4x
d.
sec2 4x
e.
cosec 4x


Solution:

(x) = tanh- 1 (sin 4x)
[Write the function.]

′(x) = ddx (tanh- 1 (sin 4x))
[Find ′(x).]

= 11-sin2 4x (4cos 4x)
[Use Chain Rule.]

= 4cos 4xcos2 4x = 4sec 4x

The derivative of (x) = tanh- 1 (sin 4x) is 4sec 4x.


Correct answer : (2)
 8.  
Find the derivative of (x) = tan- 1 5x + tanh- 1 5x.
a.
101-625x4
b.
- 101-625x4
c.
101+625x2
d.
10x21-625x4
e.
- 10x21-625x4


Solution:

(x) = tan- 1 5x + tanh- 1 5x
[Write the function.]

′ (x) = ddx (tan- 1 5x + tanh- 1 5x)
[Find ′(x).]

= 51+25x2+51-25x2
[Use the Sum Rule.]

= 5-125x2+5+125x2(1+25x2) (1-25x2)

= 101-625x4
[Simplify.]

The derivative of (x) = tan- 1 5x + tanh- 1 5x is 101-625x4


Correct answer : (1)
 9.  
Find the derivative of sinh- 1 3x + sinh- 1 4y = 4.
a.
- 4y2-13x2-1
b.
- 16x2-19y2-1
c.
- 1+3x21+4y2
d.
- 3 41-16y21-9x2
e.
- 3 41+16y21+9x2


Solution:

sinh- 1 3x + sinh- 1 4y = 4
[Write the function.]

ddx (sinh- 1 3x + sinh- 1 4y) = ddx (4)
[Find f ′(x).]

31+9x2+41+16y2dydx = 0
[Use the Sum Rule.]

41+16y2dydx = - 31+9x2
[Subtract 31+x2 from both the sides.]

dydx = - 3 / 41+16y21+9x2
[Multiply by 1+16y24 on both the sides.]

The derivative of sinh- 1 3x + sinh- 1 4y = 4 is - 3 / 41+16y21+9x2


Correct answer : (5)
 10.  
Find the derivative of g(x) = xtanh- 1 7x + 1 14 loge (1 - 49x2).
a.
tanh- 1 7x + x1-49x2
b.
11+7x
c.
tanh- 1 7x + 12(1-49x2)
d.
tanh- 1 7x - 21-49x2
e.
tanh- 1 7x


Solution:

g(x) = xtanh- 1 7x + 1 / 14 loge (1 - 49x2)
[Write the function.]

g ′(x) = ddx [xtanh- 1 7x + 1 / 14 loge (1 - 49x2)]
[Find g ′(x).]

= x (71-49x2) + tanh- 1 7x (1) + 114 (11-49x2) (- 98x)
[Use Product Rule, Sum Rule.]

= 7x1-49x2 + tanh- 1 7x - 7x1-49x2

= tanh- 1 7x
[Simplify.]

The derivative of g(x) = xtanh- 1 7x + 1 / 14loge (1 - 49x2) is tanh- 1 7x.


Correct answer : (5)

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