<s>[INST] Differentiate the Function: $y=2x \log_{10}\sqrt{x}$ $y=2x\log_{10}\sqrt{x}$
Solve using: Product Rule $\left(f(x)\cdot g(x)\right)'= f(x)\cdot\frac{d}{dx}g(x)+g(x)\cdot \frac{d}{dx}f(x)$
and $\frac{d}{dx}(\log_ax)= \frac{1}{x\ \ln\ a}$
$(2x)\cdot [\log_{10}\sqrt{x}]'+(\log_{10}\sqrt{x})\cdot [2x]'$
$y'=2x\frac{1}{\sqrt{x}\ln 10}+\log_{10}\sqrt{x}\cdot 2$
Answer in book is $y'= \frac{1}{\ln10}+\log_{10}x$
 [/INST] Take $\log_{10}x$ as $\log x$.
$$y=2x\log\sqrt{x}=x\log x=\log x^x$$
$$10^y=x^x \implies y\ln10=x\ln x \implies y=\frac{x\ln x}{ \ln10}$$
Then
$$y'=\frac{1}{\ln10}\times(\ln x + 1)=\log x + \frac{1}{\ln 10}.$$
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