Question

Derive y= log2 (3x - cos2x)

Answer 1

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Answer 2

$\displaystyle \frac{d}{dx}\left(\log_{2}(3x-\cos(2x))\right)\implies \log_{2}(u)=\frac{\ln(u)}{\ln2}\implies \\\\\frac{d}{dx}\log_{2}(3x-\cos(2x))=\frac{d}{dx}\frac{\ln(3x-\cos(2x))}{\ln(2)}\implies \\\\\frac{d}{dx}=\left(\frac{1}{\ln2}\right)\cdot (\ln(u))'\cdot u'\cdot v'\implies u=3x-\cos(2x)\ v=2x\\\\\left(\frac{1}{\ln2}\right)\cdot \frac{1}{u}\cdot 3+\sin(2x)\cdot2=\boxed{\frac{3+2\sin(2x)}{\ln2(3x-\cos(2x))}}$