6.7.3 Derivatives of Hyperbolic Functions

Question 1

Derivatives of Hyperbolic Functions

Answer 1

• To differentiate the hyperbolic functions, use their definitions.
• The derivatives of the hyperbolic functions resemble those of the trigonometric functions.

Question 2

note

Answer 2

• To determine the derivatives of the hyperbolic functions you have to differentiate the exponential expressions that define them.
• When you differentiate the expression for sinh x you produce the expression for cosh x.
• You don’t have to go back to the definitions every time. After a while you will remember them.
• Notice that the derivates of the hyperbolic functions are in some ways similar to those of the trigonometric functions. However, there are some differences.
• The derivative of cosh x is sinh x, even though the derivative of cos x is –sin x.
• And the derivative of sech x is –sech x tanh x, even though the derivative of sec x does not have a negative sign in front.
• Here is an example that looks pretty mean. However, you only need the chain rule and one of the derivatives you just learned.
• Notice that the exponent here is sinh x. That’s the inside part to which you will apply the chain rule.
• The derivative of sinh x is cosh x.
• This answer has been regrouped for a nicer presentation.

Question 3

Which of the following is the derivative of f (x) = e^xsinh (x)?

Answer 3

e^ xsinh x  (x cosh x + sinh x)

Question 4

Find the derivative of y=ln[tanh(x/4)].

Answer 4

1/2sinh(x/2)

Question 5

Which of the following is not equivalent to

d/dx(coshx)?

Answer 5

d/dx[e^x−e^−x/2]

Question 6

Find the derivative of y=(1/2)(sinh2x−2x).

Answer 6

2 sinh^2x

Question 7

To find the derivatives of some hyperbolic functions, you can use the quotient rule.Which of the following steps related to finding d/dx(tanhx) is not correct?

Answer 7

d/dx(tanhx)=sechx

Question 8

Which of the following is the derivative of

f (x) = e ^cosh (x)?

Answer 8

e^ cosh x  (sinh x)

Question 9

Which of the following is the derivative of

f (x) = ln (sinh x^3 )?

Answer 9

(3x ^2 ) (coth x ^3 )

Question 10

Which of the following is not a correct expression for d/dx[sinhx]?

Answer 10

e^x−e^−x/2

Question 11

Which of the following statements about the derivatives of hyperbolic functions is not correct?

Answer 11

d/dx[tanhx]=1/ d/dx[cothx]

Question 12

Find the derivative of y=xtanh(x/2)−x^2(coshx).

Answer 12

(x/2)sech^2(x/2)+tanh(x/2)−x^2sinhx−2xcoshx