$\"\"$

\

(a).

\

Prove that $\"\"$ .

\

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

\

$\"\"$

\

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

\

$\"\"$

\

$\"\"$

\

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

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

$\"\"$ .

\

$\"\"$

\

(b).

\

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 : $\"\"$ .

\

$\"\"$ .

\

$\"\"$ .

\

$\"\"$

\

(c).

\

Prove that $\"\"$ .

\

From reciprocal hyperbolic trigonometry : $\"\"$

\

$\"\"$ .

\

$\"\"$ .

\

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

\

$\"\"$ .

\

Since $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

(d).

\

Prove that $\"\"$ .

\

From reciprocal hyperbolic trigonometry : $\"\"$

\

$\"\"$ .

\

$\"\"$ .

\

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

\

$\"\"$ .

\

Since $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

$\"\"$

\

(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 : $\"\"$ .

\

$\"\"$ .

\

$\"\"$ .

\

$\"\"$

\

(a). $\"\"$ .

\

(b). $\"\"$ .

\

(c). $\"\"$ .

\

(d). $\"\"$ .

\

(e). $\"\"$ .