What is (\partial)/(\partial x)((x-2y^2)/(x^2-y^3)) ?

The answer to (\partial)/(\partial x)((x-2y^2)/(x^2-y^3)) is (-x^2+4xy^2-y^3)/((x^2-y^3)^2)

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(\partial)/(\partial x)((x-2y^2)/(x^2-y^3))

\frac{\partial}{\partial x} (\frac{x - 2 y ^{2}}{x ^{2} - y ^{3}})

\frac{- x ^{2} + 4 x y ^{2} - y ^{3}}{( x ^{2} - y ^{3} ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{x - 2 y ^{2}}{x ^{2} - y ^{3}})

Treat

y

as a constant

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{\partial}{\partial x} ( x - 2 y ^{2} ) ( x ^{2} - y ^{3} ) - \frac{\partial}{\partial x} ( x ^{2} - y ^{3} ) ( x - 2 y ^{2} )}{( x ^{2} - y ^{3} ) ^{2}}

\frac{\partial}{\partial x} (x - 2 y^{2}) = 1

\frac{\partial}{\partial x} (x^{2} - y^{3}) = 2 x

= \frac{1 \cdot ( x ^{2} - y ^{3} ) - 2 x ( x - 2 y ^{2} )}{( x ^{2} - y ^{3} ) ^{2}}

Simplify

\frac{1 \cdot ( x ^{2} - y ^{3} ) - 2 x ( x - 2 y ^{2} )}{( x ^{2} - y ^{3} ) ^{2}} : \frac{- x ^{2} + 4 x y ^{2} - y ^{3}}{( x ^{2} - y ^{3} ) ^{2}}

= \frac{- x ^{2} + 4 x y ^{2} - y ^{3}}{( x ^{2} - y ^{3} ) ^{2}}

What is (\partial)/(\partial x)((x-2y^2)/(x^2-y^3)) ?
The answer to (\partial)/(\partial x)((x-2y^2)/(x^2-y^3)) is (-x^2+4xy^2-y^3)/((x^2-y^3)^2)