Answer
$$ - \frac{1}{x}$$
Work Step by Step
$$\eqalign{
& y = \ln \frac{{10}}{x} \cr
& {\text{use the property of logarithms }}\ln \frac{a}{b} = \ln a - \ln b \cr
& y = \ln 10 - \ln x \cr
& {\text{Find the derivative of }}y{\text{ with respect to }}x \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {\ln 10 - \ln x} \right] \cr
& {\text{use the sum rule for derivatives}} \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {\ln 10} \right] - \frac{d}{{dx}}\left[ {\ln x} \right] \cr
& {\text{use the formula }}\frac{d}{{dx}}\ln u = \frac{1}{u}\frac{{du}}{{dx}}{\text{, }}u > 0\cr
& {\text{ where }}u{\text{ is any differentiable function of }}x \cr
& {\text{then:}} \cr
& \frac{{dy}}{{dx}} = 0 - \left( {\frac{1}{x}} \right) \cr
& {\text{solve the derivative and simplify}} \cr
& \frac{{dy}}{{dx}} = - \frac{1}{x} \cr} $$
