Chapter 3 - Section 3.8 - Derivatives of Inverse Functions and Logarithms - Exercises - Page 175: 57

$$y'=\frac{1-t}{t}$$

$$y=\ln(3t e^{-t})$$ We have $$(\ln t)'=\frac{1}{t}$$ $$(e^{t})'=e^t$$ So the derivative of $y$ is $$y'=\frac{1}{3t e^{-t}}(3t e^{-t})'=\frac{3\Big(e^{-t}+t e^{-t}(-t)'\Big)}{3t e^{-t}}$$ $$y'=\frac{e^{-t}-t e^{-t}}{t e^{-t}}=\frac{e^{-t}(1-t)}{t e^{-t}}$$ $$y'=\frac{1-t}{t}$$