Answer
$$
U_r =(\frac{-1}{r^2}-\frac{t}{r})e^{-rt},\quad
U_t= -e^{-rt}.
$$
Work Step by Step
Recall the product rule: $(uv)'=u'v+uv'$
Recall that $(e^x)'=e^x$
Since $ U=e^{-rt}/r=r^{-1}e^{-rt}$, then by using the product rule, we have
$$
U_r=-r^{-2}e^{-rt}+r^{-1}e^{-rt}(-t)=(\frac{-1}{r^2}-\frac{t}{r})e^{-rt},\\
U_t= r^{-1}e^{-rt}(-r)=-e^{-rt}.
$$
