Use logarithmic differentiation.
sqrt of 10^(5-x) is the same as 10^[(5-x)/2]
Take the natural log of both sides.
ln y = ln 10^[(5-x)/2]
ln y = [(5-x)/2] ln 10 because lnx^a = a lnx
then differentiate:
(1/y)(dy/dx) = (-1/2)ln10
solve for dy/dx:
dy/dx = (-1/2)(ln10)(10^[(5-x)/2])
edit: The answer sheet is wrong? :P