﻿ Derivatives of Hyperbolic Functions Worksheet | Problems & Solutions

Derivatives of Hyperbolic Functions Worksheet

Derivatives of Hyperbolic Functions Worksheet
• Page 1
1.
Find the derivative of $f$($x$) = 7$x$5 cosh 6$x$.
 a. 7$x$4 ( - $x$ sinh 6$x$ + 5cosh 6$x$) b. 7$x$5 (6sinh 6$x$ + $x$ cosh 6$x$) c. 7$x$4 (6$x$ sinh 6$x$ + 5cosh 6$x$) d. 7$x$4 (6$x$ sinh 6$x$ + cosh 6$x$) e. 35$x$4 sinh 6$x$

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)

2.
Find the derivative of $g$($x$) = ${e}^{8x}+{x}^{2}$ sinh 7$x$.
 a. 8 (7$x$ cosh 7$x$ + 2sinh 7$x$) b. ${e}^{8x}$ + 14$x$ cosh 7$x$ c. ${e}^{8x}$ $x$ cosh 7$x$ d. (- 7$x$ cosh 7$x$ + 2sinh 7$x$) e. (7$x$ cosh 7$x$ + sinh 7$x$)

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)

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

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)

4.
Which of the following is the derivative of ($x$) = $e$cosh2 2$x$?
 a. - $e$cosh2 2$x$ (sinh 4$x$) b. 2$e$cosh2 2$x$ (sinh 4$x$) c. 4$e$cosh2 2$x$ (cosh 2$x$) d. $e$cosh2 4$x$ e. 4$e$cosh2 2$x$ (cosh 4$x$)

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)

5.
What is the derivative of $y$ = ?
 a. b. cosh $t$ - c. cosh 9$t$ d. - e.

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

6.
Find $\frac{dy}{dx}$, if $y$ = $\frac{1}{4}$ cosh $x$ - $\frac{{x}^{2}}{9}$.
 a. $\frac{1}{4}$ sinh $x$ - $x$ b. - $\frac{1}{4}$ sinh $x$ - $x$ c. $\frac{1}{4}$ sinh $x$ - $\frac{2}{9}$$x$ d. $\frac{1}{4}$ sinh $x$ - $\frac{2}{9}$${x}^{3}$ e. $\frac{1}{4}$ sinh $x$ - $x$3

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

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

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.

8.
Find the derivative of ($x$) = tan- 1 5$x$ + tanh- 1 5$x$.
 a. $\frac{10}{1-625{x}^{4}}$ b. - $\frac{10}{1-625{x}^{4}}$ c. $\frac{10}{1+625{x}^{2}}$ d. $\frac{10{x}^{2}}{1-625{x}^{4}}$ e. - $\frac{10{x}^{2}}{1-625{x}^{4}}$

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

9.
Find the derivative of sinh- 1 3$x$ + sinh- 1 4$y$ = 4.
 a. - $\frac{4{y}^{2}-1}{3{x}^{2}-1}$ b. - $\sqrt{\frac{16{x}^{2}-1}{9{y}^{2}-1}}$ c. - $\sqrt{\frac{1+3{x}^{2}}{1+4{y}^{2}}}$ d. - $\frac{3}{4}$$\sqrt{\frac{1-16{y}^{2}}{1-9{x}^{2}}}$ e. - $\frac{3}{4}$$\sqrt{\frac{1+16{y}^{2}}{1+9{x}^{2}}}$

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

10.
Find the derivative of $g$($x$) = $x$tanh- 1 7$x$ + $\frac{1}{14}$ log$e$ (1 - 49$x$2).
 a. tanh- 1 7$x$ + $\frac{x}{1-49{x}^{2}}$ b. $\frac{1}{1+7x}$ c. tanh- 1 7$x$ + $\frac{1}{2\left(1-49{x}^{2}\right)}$ d. tanh- 1 7$x$ - $\frac{2}{1-49{x}^{2}}$ e. tanh- 1 7$x$

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.