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Hyperbolic Functions

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Hyperbolic Functions

The hyperbolic functions are defined in terms of certain combinations of $e^{x}$ and $e^{- x}$ . These functions arise naturally in various engineering and physics applications, including the study of water waves and vibrations of elastic membranes. Another common use for a hyperbolic function is the representation of a hanging chain or cable, also known as a catenary.

The shape of a strand of silk in a spider’s web can be described in terms of a hyperbolic function. The same shape applies to a chain or cable hanging from two supports with only its own weight.

The hyperbolic sine and hyperbolic cosine are defined as

$\sinh x = \frac{e^{x} - e^{- x}}{2}$ and $\cosh x = \frac{e^{x} + e^{- x}}{2}$

The other hyperbolic functions are then defined in terms of $\sinh x$ and $\cosh x$ .

Activity

Differentiation formulas for the hyperbolic functions.

$f (x)$	$\frac{d}{dx} f (x)$
$\sinh (x)$	$\cosh (x)$
$\cosh (x)$	$\sinh (x)$
$\tanh (x)$	${sech}^{2} (x)$
$\coth (x)$	$- {csch}^{2} (x)$
$sech (x)$	$- sech (x) \tanh (x)$
$csch (x)$	$- csch (x) \coth (x)$

Exercise 1

Evaluate the following derivative.

Enclose numerators and denominators in parentheses. For example, $(a - b) / (1 + n)$ . Include a multiplication sign between symbols and enclose arguments of functions in parentheses. For example, $\tanh (a * x)$ .

$\frac{d}{dx} (\tanh (x^{6} + 7 x)) =$

Hint	Penalty
Hint	0.0

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Exercise 2

Evaluate the following derivative.

$\frac{d}{dx} (\frac{1}{{(\sinh x)}^{5}}) =$

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Exercise 3

Find the derivatives of $f (x) = \tanh (\sqrt{x^{2} + 1})$ .

Include a multiplication sign between symbols. For example, $a * x$ . Enclose arguments of functions, numerators, and denominators in parentheses. For example, $\sin (a * x)$ or $(a - b) / (1 + n)$ .

$\frac{d}{dx} (\tanh (\sqrt{x^{2} + 1})) =$

Hint	Penalty
Hint	0.0

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