Hyperbolic functions
$$ 1. \ \frac {d}{dx} \sin hx = \cos hx $$
$$ 2. \ \frac {d}{dx} \cos hx = \sin hx $$
$$ 3. \ \frac {d}{dx} \tan hx = \sec h^2x $$
$$ 4. \ \frac {d}{dx} \cot hx = -cosec \ h^2x $$
$$ 5. \ \frac {d}{dx} \sec hx = -\sec hx \cdot \tan hx $$
$$ 6. \ \frac {d}{dx} cosec \ hx = -cosec \ hx \cdot \cot hx $$
Example 1:
$$ Differentiate \ y = \cos h \sqrt {x} $$
Solution:
$$ \frac {d}{dx} \left( \cos h \sqrt {x} \right) = \sin h \sqrt {x} \cdot \frac {d}{dx} \sqrt {x} $$
$$ \frac {d}{dx} \left( \cos h \sqrt {x} \right) = \frac {\sin h \sqrt {x}}{2 \sqrt {x}} $$
Example 2:
$$ Differentiate \ f(x) = \tan h(4x) $$
Solution:
$$ f'(x) = \sec h^2(4x) \cdot [4x]’ $$
$$ f'(x) = \sec h^2(4x) \cdot 4 $$
$$ f'(x) = 4 \sec h^2(4x) $$