Chapter 3 - Section 3.8 - Derivatives of Inverse Functions and Logarithms - Exercises - Page 175: 37

$$y'=\frac{\tan(\ln\theta)}{\theta}$$

$$y=\ln(\sec(\ln\theta))$$ The derivative of y: $$y'=\frac{1}{\sec(\ln\theta)}\times(\sec(\ln \theta))'=\frac{1}{\sec(\ln\theta)}\times(\sec(\ln \theta)\tan(\ln \theta))\times(\ln \theta)'$$ $$y'=\frac{1}{\sec(\ln\theta)}\times(\sec(\ln \theta)\tan(\ln \theta))\times\frac{1}{\theta}$$ $$y'=\frac{\sec(\ln\theta)\tan(\ln\theta)}{\theta\sec(\ln\theta)}$$ $$y'=\frac{\tan(\ln\theta)}{\theta}$$