$y=2\sqrt{ x }$
Take the natural logarithm of both sides of the equation$\ln\left({y}\right)=\ln\left({2\sqrt{ x }}\right)$
Simplify the expression$\ln\left({y}\right)=\ln\left({2}\right)+\frac{ 1 }{ 2 } \times \ln\left({x}\right)$
Differentiate both sides of the equation implicitly with respect to $x$$\frac{ \mathrm{d} }{ \mathrm{d}x} \left( \ln\left({y}\right) \right)=\frac{ \mathrm{d} }{ \mathrm{d}x} \left( \ln\left({2}\right)+\frac{ 1 }{ 2 } \times \ln\left({x}\right) \right)$
Use the chain rule $\frac{ \mathrm{d} }{ \mathrm{d}x} \left( \ln\left({y}\right) \right)=\frac{ \mathrm{d} }{ \mathrm{d}y} \left( \ln\left({y}\right) \right) \times \frac{ \mathrm{d}y }{ \mathrm{d}y }$ to find the derivative$\frac{ \mathrm{d} }{ \mathrm{d}y} \left( \ln\left({y}\right) \right) \times \frac{ \mathrm{d}y }{ \mathrm{d}y }=\frac{ \mathrm{d} }{ \mathrm{d}x} \left( \ln\left({2}\right)+\frac{ 1 }{ 2 } \times \ln\left({x}\right) \right)$
Use differentiation rule $\frac{ \mathrm{d} }{ \mathrm{d}x} \left( f+g \right)=\frac{ \mathrm{d} }{ \mathrm{d}x} \left( f \right)+\frac{ \mathrm{d} }{ \mathrm{d}x} \left( g \right)$$\frac{ \mathrm{d} }{ \mathrm{d}y} \left( \ln\left({y}\right) \right) \times \frac{ \mathrm{d}y }{ \mathrm{d}y }=\frac{ \mathrm{d} }{ \mathrm{d}x} \left( \ln\left({2}\right) \right)+\frac{ \mathrm{d} }{ \mathrm{d}x} \left( \frac{ 1 }{ 2 } \times \ln\left({x}\right) \right)$
Use $\frac{ \mathrm{d} }{ \mathrm{d}x} \left( \ln\left({x}\right) \right)=\frac{ 1 }{ x }$ to find derivative$\frac{ 1 }{ y } \times \frac{ \mathrm{d}y }{ \mathrm{d}y }=\frac{ \mathrm{d} }{ \mathrm{d}x} \left( \ln\left({2}\right) \right)+\frac{ \mathrm{d} }{ \mathrm{d}x} \left( \frac{ 1 }{ 2 } \times \ln\left({x}\right) \right)$
Find the derivative$\frac{ 1 }{ y } \times \frac{ \mathrm{d}y }{ \mathrm{d}y }=0+\frac{ \mathrm{d} }{ \mathrm{d}x} \left( \frac{ 1 }{ 2 } \times \ln\left({x}\right) \right)$
Find the derivative$\frac{ 1 }{ y } \times \frac{ \mathrm{d}y }{ \mathrm{d}y }=0+\frac{ 1 }{ 2 } \times \frac{ 1 }{ x }$
Simplify the expression$\frac{ 1 }{ y } \times \frac{ \mathrm{d}y }{ \mathrm{d}y }=\frac{ 1 }{ 2x }$
Multiply both sides of the equation by $y$$\frac{ \mathrm{d}y }{ \mathrm{d}y }=y \times \frac{ 1 }{ 2x }$
Substitute the initial equation $y=2\sqrt{ x }$ to express the derivative only in terms of $x$$\frac{ \mathrm{d}y }{ \mathrm{d}y }=2\sqrt{ x } \times \frac{ 1 }{ 2x }$
Simplify the expression$\frac{ \mathrm{d}y }{ \mathrm{d}y }=\frac{ \sqrt{ x } }{ x }$