Answer
$$f'\left( x \right) = \frac{{10.8{x^3} + 90{x^2} - 8.1}}{{{{\left( {2.7x + 15} \right)}^2}}}$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = \frac{{2{x^3} + 3}}{{2.7x + 15}} \cr
& {\text{Calculate the derivative of the function}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left( {\frac{{2{x^3} + 3}}{{2.7x + 15}}} \right) \cr
& {\text{Use the quotient rule }}\left( {{\text{see the page 237}}} \right) \cr
& f'\left( x \right) = \frac{{\left( {2.7x + 15} \right)\left( {2{x^3} + 3} \right)' - \left( {2{x^3} + 3} \right)\left( {2.7x + 15} \right)'}}{{{{\left( {2.7x + 15} \right)}^2}}} \cr
& {\text{compute derivatives}} \cr
& f'\left( x \right) = \frac{{\left( {2.7x + 15} \right)\left( {6{x^2}} \right) - \left( {2{x^3} + 3} \right)\left( {2.7} \right)}}{{{{\left( {2.7x + 15} \right)}^2}}} \cr
& {\text{Multiplying}} \cr
& f'\left( x \right) = \frac{{16.2{x^3} + 90{x^2} - 5.4{x^3} - 8.1}}{{{{\left( {2.7x + 15} \right)}^2}}} \cr
& f'\left( x \right) = \frac{{10.8{x^3} + 90{x^2} - 8.1}}{{{{\left( {2.7x + 15} \right)}^2}}} \cr} $$
