$\"\"$

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(a).

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Prove that $\"\"$ .

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The definition of hyperbolic trigonometry : $\"\"$ and $\"\"$ .

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$\"\"$

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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The exponential form of derivative: $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$ .

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$\"\"$

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(b).

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Prove that $\"\"$ .

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The definition of hyperbolic trigonometry : $\"\"$ and $\"\"$ .

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$\"\"$

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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The quitent rule of derivative : $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Since the hyperbolic identity : $\"\"$ .

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$\"\"$ .

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$\"\"$ .

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$\"\"$

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(c).

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Prove that $\"\"$ .

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From reciprocal hyperbolic trigonometry : $\"\"$ .

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$\"\"$

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$\"\"$

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The power rule of derivative : $\"\"$ .

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$\"\"$ .

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Since $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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(d).

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Prove that $\"\"$ .

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From reciprocal hyperbolic trigonometry : $\"\"$ .

\

$\"\"$

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$\"\"$

\

The power rule of derivative : $\"\"$ .

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$\"\"$ .

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Since $\"\"$ .

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$\"\"$

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$\"\"$

\

$\"\"$

\

$\"\"$ .

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$\"\"$

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(e).

\

Prove that $\"\"$ .

\

The definition of hyperbolic trigonometry: $\"\"$ and $\"\"$ .

\

$\"\"$

\

Apply derivative on each side with respect to $\"\"$ .

\

$\"\"$

\

The quitent rule of derivative : $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

Since the hyperbolic identity : $\"\"$ .

\

$\"\"$

\

$\"\"$ .

\

$\"\"$

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(a). $\"\"$ .

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(b). $\"\"$ .

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(c). $\"\"$ .

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(d). $\"\"$ .

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(e). $\"\"$ .