# Solve for: f(x)=2sqrt(x)

## Expression: $f\left( x \right)=2\sqrt{ x }$

Substitute $y$ for $f\left( x \right)$ into the equation

$y=2\sqrt{ x }$

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

$\frac{ \mathrm{d}y }{ \mathrm{d}y }=\frac{ \sqrt{ x } }{ x }$

Random Posts
Random Articles