Chapter 11 - Section 11.5 - Derivatives of Logarithmic and Exponential Functions - Exercises - Page 842: 39

$ \displaystyle \frac{1-t^{-2}}{(t+t^{-1})\ln 3}$

Apply $\displaystyle \frac{d}{dx}[\log_{b}u]=\frac{\mathrm{l}}{u\ln b}\frac{du}{dx} \quad $(see p. 836) $u(t)=t+t^{-1}$ $\displaystyle \frac{du}{dt}=1-t^{-2}.$ $\displaystyle \frac{d}{dt}[\log_{3}(t+t^{-1})]=\frac{\mathrm{l}}{(t+t^{-1})\ln 3}\cdot(1-t^{-2}.)$ $=\displaystyle \frac{1-t^{-2}}{(t+t^{-1})\ln 3}$