For y=1/f(x)
dy/dx = -f'(x) /[f(x)]^2 (denominator is strictly positive, so its consideration regarding the sign of dy/dx is superfluous)
so we have: the sign of dy/dx = the opposite sign of f'(x). What does this tell us? When f(x) is decreasing, 1/f(x) will be increasing and vice versa...