Differentiation of Inverse Hyperbolic Functions

Example 1:

$$ y=\sinh ^{-1} 8x,\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$

Solution:

$$ y=\sinh ^{-1} 8x$$ $$ \frac{dy}{dx}=\frac{d}{dx}\sinh ^{-1} 8x$$ $$ \textup{Let}\; u= 8x$$ $$ \frac{dy}{dx}=\frac{d}{dx} \sinh ^{-1} u$$ $$ \frac{dy}{dx}=\frac{d}{du} \sinh ^{-1} u \frac{du}{dx}$$ $$ \because\frac{d}{dx}\sinh ^{-1} x=\frac{1}{\sqrt{x^2+1}}$$ $$ \therefore\frac{dy}{dx}=\frac{1}{\sqrt{u^2+1}} \frac{d}{dx}8x$$ $$ \frac{dy}{dx}=\frac{1}{\sqrt{(8x)^2+1}}(8\frac{d}{dx}x)$$ $$ \because\frac{d}{dx} x=1 $$ $$ \therefore \frac{dy}{dx}=\frac{1}{\sqrt{64x^2+1}}(8(1))$$ $$ \frac{dy}{dx}=\frac{1}{\sqrt{64x^2+1}} (8)$$ $$ \frac{dy}{dx}=\frac{8}{\sqrt{64x^2+1}}$$