Answer
$$y' = \frac{{15{x^2}{e^{\sqrt {1 + 5{x^3}} }}}}{{2{{\left( {1 + 5{x^3}} \right)}^{1/2}}}}$$
Work Step by Step
$$\eqalign{
& y = {e^{\sqrt {1 + 5{x^3}} }} \cr
& {\text{use the chain rule}} \cr
& y' = {e^{\sqrt {1 + 5{x^3}} }}\left( {\sqrt {1 + 5{x^3}} } \right)' \cr
& {\text{radical properties}} \cr
& y' = {e^{\sqrt {1 + 5{x^3}} }}\left( {{{\left( {1 + 5{x^3}} \right)}^{1/2}}} \right)' \cr
& {\text{product rule}} \cr
& y' = {e^{\sqrt {1 + 5{x^3}} }}\left( {\frac{1}{2}{{\left( {1 + 5{x^3}} \right)}^{ - 1/2}}} \right)\left( {1 + 5{x^3}} \right)' \cr
& y' = {e^{\sqrt {1 + 5{x^3}} }}\left( {\frac{1}{2}{{\left( {1 + 5{x^3}} \right)}^{ - 1/2}}} \right)\left( {15{x^2}} \right) \cr
& {\text{simplify}} \cr
& y' = \frac{{15{x^2}{e^{\sqrt {1 + 5{x^3}} }}}}{{2{{\left( {1 + 5{x^3}} \right)}^{1/2}}}} \cr} $$
