Answer
$f(x)=-2\ln x+3x-2$
Work Step by Step
$f^{\prime}(x)=\displaystyle \int\frac{2}{x^{2}}dx=2\int x^{-2}dx\\=2\dfrac{x^{-1}}{-1}+C_{1}=-\dfrac{2}{x}+C_{1}$
since $f^{\prime}(1)=1,$
$1=-\displaystyle \frac{2}{1}+C_{1}$
$C_{1}=3\displaystyle \Rightarrow f^{\prime}(x)=-\frac{2}{x}+3$
$f(x)=\displaystyle \int(-\frac{2}{x}+3)dx=-2\int\frac{dx}{x}+3\int dx$
$f(x)=-2\ln|x|+3x+C$
since $f(1)=1, $and $x>0\Rightarrow|x|=x,$
$1=-2\ln 1+3\cdot 1+C$
$C=-2$
$f(x)=-2\ln x+3x-2$
