Solve for: y=\sqrt[3]{5x}

Expression: $y=\sqrt[3]{5x}$

Take the derivative of both sides

$y '=\frac{ \mathrm{d} }{ \mathrm{d}x} \left( \sqrt[3]{5x} \right)$

Use $\sqrt[n]{{a}^{m}}={a}^{\frac{ m }{ n }}$ to transform the expression

$y '=\frac{ \mathrm{d} }{ \mathrm{d}x} \left( {\left( 5x \right)}^{\frac{ 1 }{ 3 }} \right)$

To raise a product to a power, raise each factor to that power

$y '=\frac{ \mathrm{d} }{ \mathrm{d}x} \left( {5}^{\frac{ 1 }{ 3 }} \times {x}^{\frac{ 1 }{ 3 }} \right)$

Use differentiation rule $\frac{ \mathrm{d} }{ \mathrm{d}x} \left( a \times f \right)=a \times \frac{ \mathrm{d} }{ \mathrm{d}x} \left( f \right)$

$y '={5}^{\frac{ 1 }{ 3 }} \times \frac{ \mathrm{d} }{ \mathrm{d}x} \left( {x}^{\frac{ 1 }{ 3 }} \right)$

Use $\frac{ \mathrm{d} }{ \mathrm{d}x} \left( {x}^{n} \right)=n \times {x}^{n-1}$ to find derivative

$y '={5}^{\frac{ 1 }{ 3 }} \times \frac{ 1 }{ 3 }{x}^{-\frac{ 2 }{ 3 }}$

Simplify the expression

$y '=\frac{ \sqrt[3]{5} }{ 3\sqrt[3]{{x}^{2}} }$