Important question- is "log x" the common logarithm (base 10) or the natural logarithm (base e)? Normally, for a calculus problem I would assume natural logarithm, but the presence of "10^x" makes me ask.
I presume you know that the derivative of -10x is -10.
10^x log x is a product so use the product rule: (10^x log x)'= (10^x)' log x+ 10^x (log x)'.
The derivative of 10^x is (10^x)'= (10^x) ln x and the derivative of log x is
1) 1/(x ln(10)) if it is common logarithm.
2) 1/x, if it is natural logarithm.