derivative of complex fcn

Question

anonymous · Answer

$$y=2x \log_{10}\sqrt{x}$$ find $$y'$$

anonymous · Answer

product rule for this one 
the derivative of $2x$ is $2$ and the derivative of $\log_{10}(x)$ is $\frac{1}{x\ln(10)}$

anonymous · Answer

whoa you have $\log(\sqrt{x})$ wasn't paying attention.
no matter, start by writing $\log(\sqrt{x})=\frac{1}{2}\log(x)$ an differentiate 
$$x\log(x)$$ again using the produce rule

anonymous · Answer

use $(fg)'=f'g+g'f$ with $f(x)=x, f'(x)=1,g(x)=\log(x), g'(x)=\frac{1}{x\ln(10)}$

anonymous · Answer

shouldnt this be chain rule?

raden · Answer

i think here, it doesnt need chain rule but product rule
satellite is right

anonymous · Answer

how is $$\log_{10}(\sqrt{x})=\frac{ 1 }{ 2 } \log_{10} x$$  ?

raden · Answer

because : 
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raden · Answer

and we use property of logarithm :
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