Chapter 6 - Inverse Functions - 6.2 Exponential Functions and Their Derivatives - 6.2 Exercises - Page 418: 38

$-\frac{(t+2)^2}{t^2e^t}$

$\frac{d}{dt}\frac{4+t}{te^t}$ $=\frac{te^t\frac{d}{dt}(4+t)-(4+t)\frac{d}{dt}te^t}{(te^t)^2}$ $=\frac{te^t*1-(4+t)(t\frac{d}{dt}e^t+e^t\frac{d}{dt}t)}{(te^t)^2}$ $=\frac{te^t-(4+t)(te^t+e^t)}{(te^t)^2}$ $=\frac{te^t-(4+t)(t+1)e^t}{(te^t)^2}$ $=\frac{(t-(5t+4+t^2))e^t}{(te^t)^2}$ $=\frac{-(t^2+4t+4)}{t^2e^t}$ $=-\frac{(t+2)^2}{t^2e^t}$