Answer
$$f'\left( x \right) = {e^{15.7{x^3}}}\left( {\frac{1}{x} + 15.7{x^2}\ln \left( {15.7{x^3}} \right)} \right)$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = \left[ {\ln \left( {15.7{x^3}} \right)} \right]\left( {{e^{15.7{x^3}}}} \right) \cr
& {\text{calculate the derivative of the function}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left( {\left[ {\ln \left( {15.7{x^3}} \right)} \right]\left( {{e^{15.7{x^3}}}} \right)} \right) \cr
& {\text{use product rule}} \cr
& f'\left( x \right) = \left( {{e^{15.7{x^3}}}} \right)\frac{d}{{dx}}\left( {\left[ {\ln \left( {15.7{x^3}} \right)} \right]} \right) + \left( {\left[ {\ln \left( {15.7{x^3}} \right)} \right]} \right)\frac{d}{{dx}}\left( {{e^{15.7{x^3}}}} \right) \cr
& {\text{compute derivatives}} \cr
& f'\left( x \right) = \left( {{e^{15.7{x^3}}}} \right)\left( {\frac{{15.7\left( {3{x^2}} \right)}}{{15.7{x^3}}}} \right) + \ln \left( {15.7{x^3}} \right)\left( {15.7} \right)\left( {3{x^2}} \right)\left( {{e^{15.7{x^3}}}} \right) \cr
& {\text{simplifying}} \cr
& f'\left( x \right) = \left( {{e^{15.7{x^3}}}} \right)\left( {\frac{3}{x}} \right) + 15.7\ln \left( {15.7{x^3}} \right)\left( {3{x^2}} \right)\left( {{e^{15.7{x^3}}}} \right) \cr
& {\text{factor out 3}}{e^{15.7{x^3}}} \cr
& f'\left( x \right) = {e^{15.7{x^3}}}\left( {\frac{1}{x} + 15.7{x^2}\ln \left( {15.7{x^3}} \right)} \right) \cr} $$
