6.7.3 Derivatives of Hyperbolic Functions Flashcards Preview

AP Calculus AB > 6.7.3 Derivatives of Hyperbolic Functions > Flashcards

Flashcards in 6.7.3 Derivatives of Hyperbolic Functions Deck (12):
1

Derivatives of Hyperbolic Functions

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

2

note

- 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.

3

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

e^ xsinh x  (x cosh x + sinh x)

4

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

1/2sinh(x/2)

5

Which of the following is not equivalent to
d/dx(coshx)?

d/dx[e^x−e^−x/2]

6

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

2 sinh^2x

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?

d/dx(tanhx)=sechx

8

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

e^ cosh x  (sinh x)

9

Which of the following is the derivative of
f (x) = ln (sinh x^3 )?

(3x ^2 ) (coth x ^3 )

10

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

e^x−e^−x/2

11

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

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

12

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

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

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