#### Answer

$$3$$

#### Work Step by Step

$$\eqalign{
& {\left. {\frac{d}{{dx}}\left( {x\ln {x^3}} \right)} \right|_{x = 1}} \cr
& {\text{use product rule}} \cr
& = x\frac{d}{{dx}}\left( {\ln {x^3}} \right) + \ln {x^3}\frac{d}{{dx}}\left( x \right) \cr
& = x\left( {\frac{{3{x^2}}}{{{x^3}}}} \right) + \ln {x^3}\left( 1 \right) \cr
& = 3 + \ln {x^3} \cr
& {\text{evaluate the derivative at }}x = 1 \cr
& = 3 + \ln {\left( 1 \right)^3} \cr
& {\text{Simplify}} \cr
& = 3 \cr} $$