$f '\left( x \right)=\frac{ \mathrm{d} }{ \mathrm{d}x} \left( \frac{ 2{e}^{4x} }{ \sqrt{ x } } \right)$
$f '\left( x \right)=\frac{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( 2{e}^{4x} \right) \times \sqrt{ x }-2{e}^{4x} \times \frac{ \mathrm{d} }{ \mathrm{d}x} \left( \sqrt{ x } \right) }{ {\sqrt{ x }}^{2} }$
$f '\left( x \right)=\frac{ 2{e}^{4x} \times 4\sqrt{ x }-2{e}^{4x} \times \frac{ \mathrm{d} }{ \mathrm{d}x} \left( \sqrt{ x } \right) }{ {\sqrt{ x }}^{2} }$
$f '\left( x \right)=\frac{ 2{e}^{4x} \times 4\sqrt{ x }-2{e}^{4x} \times \frac{ 1 }{ 2\sqrt{ x } } }{ {\sqrt{ x }}^{2} }$
$f '\left( x \right)=\frac{ 8x{e}^{4x}-{e}^{4x} }{ \sqrt{ x }x }$