<s>[INST] Derivative of $(2x-1)(x+3)^{\frac{1}{2}}$ Find the derivative of 
$(2x-1)(x+3)^{\frac{1}{2}}$ 
My try - 
$(2x-1)(\frac{1}{2} (x+3)^{\frac{-1}{2}} (x+0) + (x+3)^{\frac{1}{2}} (2)$ 
$ = (x+3)^{\frac{-1}{2}} (\frac{1}{2}(2x-1) + 2(x+3) $ 
$= \frac{3x+5.5}{2 (x+3)^{\frac{1}{2}}} $ 
My numerator is wrong and should be 
$6x+11$ . Where did I go wrong ? Thanks !! 
 [/INST] Using production rule and rule for derivative of complicated function:
$$(f\cdot g)' = f'\cdot g + f \cdot g' \\ f(\varphi(x))' = f' \cdot (\varphi(x))'$$
We have:
$$(2x-1)'\cdot(x+3)^{\frac{1}{2}}+(2x-1)\cdot((x+3)^{\frac{1}{2}})'\cdot(x+3)'$$
$$(2x-1)' = 2 \\ ((x+3)^{\frac{1}{2}})' = \frac{1}{2}\cdot(x+3)^{\frac{-1}{2}} \\ (x+3)' = 1$$
You can do the rest I think :)
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