Derivative of $10^x\cdot\log_{10}(x)$ 
Derive  $10^x\cdot\log_{10}(x)$ 

$$10^x\cdot \ln(10)\cdot \log_{10}(x)+\frac{1}{x\cdot \ln(10)}\cdot 10^x$$
But WolframAlpha gives another solution. Where am I wrong?

Your answer is correct, but you can simplify it further using the change of base formula. Where $\log x$ is in base $10$ and $\ln x$ is the natural log, you have
$$10^x\cdot \ln10\cdot \log x+\frac{1}{x\cdot \ln10}\cdot 10^x$$
$$=10^x\cdot \left(\ln10\cdot \color{green}{\log x}+\frac{1}{x\cdot \ln 10}\right)$$
$$=10^x\cdot \left(\ln10\cdot \color{green}{\frac{\ln x}{\ln 10}}+\frac{1}{x\cdot \ln 10}\right)$$
$$=10^x\cdot \left(\ln x+\frac{1}{x\cdot \ln 10}\right)$$
