Question

Find dy/dx, if y=x ln (xy)​

Answer 1

$\bigstar$ Answer:

$y=xln(xy)\\\\=> \frac{dy}{dx}=ln(xy)\frac{d}{dx}(x)+x\frac{d}{dx}(ln(xy))\\\\=>\frac{dy}{dx}=ln(xy)+x*\frac{1}{xy}*[x(\frac{dy}{dx}) +y\frac{dx}{dx}]\\\\=>\frac{dy}{dx}=lnxy+\frac{1}{y}[\frac{dy}{dx}+y]\\\\=>\frac{dy}{dx} =lnxy+\frac{1}{y} .\frac{dy}{dx} +1\\\\=>\frac{dy}{dx} -\frac{x}{y} .\frac{dy}{dx} =lnxy+1\\\\=>\frac{dy}{dx} (1-\frac{x}{y} )=lnxy+1\\\\=>\frac{dy}{dx} =\frac{lnxy+1}{\frac{y-x}{y} } \\\\=>\frac{dy}{dx}=\frac{y(lnxy+1)}{y-x}$

$\bigstar$ More Information : The first idea to solve this that springs to my mind is, of course, to apply implicit differentiation, but this is not an obvious function and so I got stuck. I simply don't know how to tackle this. Because, if I take the derivative with respect to x of both sides, I get above.

>>> $\bigstar$ $\bigstar$ $\bigstar$ Hope Helps You $\bigstar$ $\bigstar$ $\bigstar$  <<<