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AP Calculus AB: 6.7.3 Derivatives of Hyperbolic Functions

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This content explains how to differentiate hyperbolic functions using their exponential definitions and the chain rule. It highlights similarities and differences between derivatives of hyperbolic and trigonometric functions, and includes example problems to illustrate the process.

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Derivatives of Hyperbolic Functions

  • To differentiate the hyperbolic functions, use their definitions.

  • The derivatives of the hyperbolic functions resemble those of the trigonometric functions.

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Key Terms

Term
Definition

Derivatives of Hyperbolic Functions

  • To differentiate the hyperbolic functions, use their definitions.

  • The derivatives of the hyperbolic functions resemble those...

note

  • To determine the derivatives of the hyperbolic functions you have to differentiate the exponential expressions that define them.

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

e^ xsinh x  (x cosh x + sinh x)

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

1/2sinh(x/2)

Which of the following is not equivalent to

d/dx(coshx)?

d/dx[e^x−e^−x/2]

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

2 sinh^2x

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AP Calculus AB: 6.7.3 Derivatives of Hyperbolic Functions
TermDefinition

Derivatives of Hyperbolic Functions

  • To differentiate the hyperbolic functions, use their definitions.

  • The derivatives of the hyperbolic functions resemble those of the trigonometric functions.

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.

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

e^ xsinh x  (x cosh x + sinh x)

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

1/2sinh(x/2)

Which of the following is not equivalent to

d/dx(coshx)?

d/dx[e^x−e^−x/2]

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

2 sinh^2x

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

Which of the following is the derivative of

f (x) = e ^cosh (x)?

e^ cosh x  (sinh x)

Which of the following is the derivative of

f (x) = ln (sinh x^3 )?

(3x ^2 ) (coth x ^3 )

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

e^x−e^−x/2

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

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

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

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