Question

Find the derivative of y with respect to x. y= ((lnx)/(2+lnx))

Answer 1

Using the known theorem
`\left((f)/(g)\right)'=(f'g-g'f)/(g^2)` you get

`y'=\left((\ln x)/(2+\ln x)\right) '=((\ln x)'(2+\ln x)-(2+\ln x)'\ln x)/((2+\ln x)^2)=((1)/(x)(2+\ln x)-(1)/(x)\ln x)/((2+\ln x)^2)`.

Therefore,
`y'=((2+\ln x)/(x)-(\ln x)/(x))/((2+\ln x)^2)=((2)/(x))/((2+\ln x)^2)=(2)/(x(2+\ln x)^2)`.