<s>[INST] 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?
 [/INST] Use the facts

$$(a^x)' = a^x \ln a, \quad (\log_a(x))' = \frac{1}{x\ln a}$$
  for $a > 0$ and $a \neq 1$.

Then use the multiplication formula for differentiation to get
\begin{align}
& (10^x \log_{10}x)' \\
= & (10^x)'\log_{10}x + 10^x (\log_{10}x)' \\
= & 10^x \ln (10) \log_{10}x + 10^x \frac{1}{x\ln 10}.
\end{align}
If you want to express your final result in natural logs, then you can simplify by writing 
$$\log_{10}x = \frac{\ln x}{\ln 10},$$
in this way the answer would agree with the wolfram output exactly.
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