Answer
$${\left( {\frac{1}{x}} \right)^x}\left( {\ln \left( {\frac{1}{x}} \right) - 1} \right)$$
Work Step by Step
$$\eqalign{
& \frac{d}{{dx}}\left( {{{\left( {\frac{1}{x}} \right)}^x}} \right) \cr
& {\text{we use the inverse relationship }}{e^{\ln x}} = x \cr
& = {e^{\ln {{\left( {\frac{1}{x}} \right)}^x}}} \cr
& = {e^{x\ln \left( {\frac{1}{x}} \right)}} \cr
& {\text{evaluate the derivative}} \cr
& = \frac{d}{{dx}}\left( {{e^{x\ln \left( {\frac{1}{x}} \right)}}} \right) \cr
& {\text{by }}\frac{d}{{dx}}\left( {{e^{u\left( x \right)}}} \right) = {e^{u\left( x \right)}}u'\left( x \right) \cr
& = {e^{x\ln \left( {\frac{1}{x}} \right)}}\frac{d}{{dx}}\left( {x\ln \left( {\frac{1}{x}} \right)} \right) \cr
& {\text{product rule}} \cr
& = {e^{x\ln \left( {\frac{1}{x}} \right)}}\left( {x\left( { - \frac{1}{x}} \right) + \ln \left( {\frac{1}{x}} \right)} \right) \cr
& {\text{simplify}} \cr
& = {\left( {\frac{1}{x}} \right)^x}\left( {\ln \left( {\frac{1}{x}} \right) - 1} \right) \cr} $$
