Prove the following formula for the derivative of hyperbolic cosine,

*Proof.* We can compute from the definition of hyperbolic cosine in terms of exponentials,

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Stumbling Robot

A Fraction of a Dot
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Tag: Hyperbolic functions

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Prove that the derivative of *cosh x* is *sinh x*

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Prove that the derivative of *sinh x* is *cosh x*

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Calculate *tanh (2x)* given *tanh x = 3/4*

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Calculate *cosh (x+y)* given *sinh x = 4/3* and *sinh y = 3/4*

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Calculate *sinh x* and *cosh x* given *tanh x = 5/13*

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Calculate *sinh x* given *cosh x = 5/4* and * x > 0*

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Calculate *cosh x* when *sinh x = 4/3*

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Prove that *coth*^{2} x – csch^{2} x = 1

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Prove that * tanh*^{2} x + sech^{2}x = 1

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Prove that *2 cosh*^{2} (x/2) = cosh x + 1

Prove the following formula for the derivative of hyperbolic cosine,

*Proof.* We can compute from the definition of hyperbolic cosine in terms of exponentials,

Prove the following formula for the derivative of the hyperbolic sine,

*Proof.* We can compute the derivative from the definition of the hyperbolic sine in terms of exponentials,

Find the value of given that .

We recall the following formulas (from Section 6.19, Exercises #7 and #8),

Then using the definition of hyperbolic tangent we compute

Find the value of given that and .

We use the identity in both cases.

Then we recall (Section 6.19, Exercise #6) the formula for ,

Therefore, we can compute

Find the values of and given that .

Since (see here, Section 6.19 Exercise #14) we have

Then, to compute we have,

Find the value of given that and .

We know . So, if we have

Thus, if ,

Find the value of when given .

We know . So, if then we have

Hence,

Prove the following identity,

* Proof. * From the definitions of hyperbolic cotangent and hyperbolic cosecant we have,

Prove the following identity,

From the definitions of hyperbolic tangent and hyperbolic secant we have,

Prove the following identity,

* Proof. * Computing directly from the definition of the hyperbolic cosine,