Chapter 5 - Logarithmic, Exponential, and Other Transcendental Functions - 5.8 Exercises - Page 390: 27

$f^{\prime}(x)=\coth x$

Apply the Chain Rule and Th. 5.18: Let $u$ be a differentiable function of $x$. $\displaystyle \frac{d}{dx}[\sinh u]=(\cosh u)u^{\prime}$ --- $u(v(x))=\ln(\sinh x)$ $f^{\prime}(x)=\displaystyle \frac{d}{dx}[u(v(x))]=\frac{du}{dv}\cdot\frac{dv}{dx}$ $=\displaystyle \frac{1}{\sinh x}(\cosh x)$ $=\displaystyle \frac{\cosh x}{\sinh x}=\coth x$