Chapter 5 - Logarithmic, Exponential, and Other Transcendental Functions - 5.8 Exercises - Page 390: 31

$f^{\prime}(t)= {\rm sech} \ t$

Apply the Chain Rule and Th. 5.18:$\displaystyle \qquad \frac{d}{dt}[\sinh u]=(\cosh u)u^{\prime}$ $\displaystyle \frac{d}{du}[\arctan u]=\frac{1}{1+u^{2}}$ $f^{\prime}(t)=\displaystyle \frac{1}{1+\sinh^{2}t}(\cosh t)$ $=\displaystyle \frac{\cosh t}{\cosh^{2}t}$ $=\displaystyle \frac{1}{\cosh t}= {\rm sech} \ t$