Differentiation of Hyperbolic Functions

Example 1:

$$ y=\sinh 5x,\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$

Solution:

$$ y=\sinh 5x$$ $$ \frac{dy}{dx}=\frac{d}{dx}\sinh 5x$$ $$ \textup{Let}\; u= 5x$$ $$ \frac{dy}{dx}=\frac{d}{dx} \sinh u$$ $$ \frac{dy}{dx}=\frac{d}{du} \sinh u \frac{du}{dx}$$ $$ \because\frac{d}{dx}\sinh x=\cosh x$$ $$ \therefore\frac{dy}{dx}=\cosh u \frac{d}{dx}4x$$ $$ \frac{dy}{dx}=\cosh 5x (5\frac{d}{dx}x)$$ $$ \because\frac{d}{dx} x=1 $$ $$ \therefore \frac{dy}{dx}=\cosh 5x (5(1))$$ $$ \frac{dy}{dx}=\cosh 5x (5)$$ $$ \frac{dy}{dx}=5\cosh 5x $$