Solve frac{1/sqrt{y}}{frac{sqrt[3]{y^2}}{y^2-y^3}}

$$\frac{ \frac{ 1 }{ \sqrt{ y } } }{ \frac{ \sqrt[3]{ { y }^{ 2 } } }{ { y }^{ 2 } - { y }^{ 3 } } }$$

Evaluate

$$y^{\frac{5}{6}}\left(1-y\right)$$

Short Solution Steps

Divide \frac{1}{\sqrt{y}} by \frac{\sqrt[3]{y^{2}}}{y^{2}-y^{3}} by multiplying \frac{1}{\sqrt{y}} by the reciprocal of \frac{\sqrt[3]{y^{2}}}{y^{2}-y^{3}}.

$$\frac{y^{2}-y^{3}}{\sqrt{y}\sqrt[3]{y^{2}}}$$

Factor the expressions that are not already factored.

$$\frac{\left(-y+1\right)y^{2}}{\sqrt{y}\sqrt[3]{y^{2}}}$$

Cancel out \sqrt{y} in both numerator and denominator.

$$\frac{\left(-y+1\right)y^{\frac{3}{2}}}{\sqrt[3]{y^{2}}}$$

Expand the expression.

$$\frac{-y^{\frac{5}{2}}+y^{\frac{3}{2}}}{y^{\frac{2}{3}}}$$

Factor the expressions that are not already factored.

$$\frac{\left(-y+1\right)y^{\frac{3}{2}}}{y^{\frac{2}{3}}}$$

Cancel out y^{\frac{2}{3}} in both numerator and denominator.

$$y^{\frac{5}{6}}\left(-y+1\right)$$

Expand the expression.

$$-y^{\frac{11}{6}}+y^{\frac{5}{6}}$$

Differentiate w.r.t. y

$$\frac{5-11y}{6\sqrt[6]{y}}$$