What is (\partial)/(\partial y)((xy^2)/(x^2y^3+1)) ?

The answer to (\partial)/(\partial y)((xy^2)/(x^2y^3+1)) is (x(-x^2y^4+2y))/((x^2y^3+1)^2)

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(\partial)/(\partial y)((xy^2)/(x^2y^3+1))

\frac{\partial}{\partial y} (\frac{x y ^{2}}{x ^{2} y ^{3} + 1})

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

Solution steps

\frac{\partial}{\partial y} (\frac{x y ^{2}}{x ^{2} y ^{3} + 1})

Treat

x

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= x \frac{\partial}{\partial y} (\frac{y ^{2}}{x ^{2} y ^{3} + 1})

Apply the Quotient Rule:

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

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

\frac{\partial}{\partial y} (y^{2}) = 2 y

\frac{\partial}{\partial y} (x^{2} y^{3} + 1) = 3 x^{2} y^{2}

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

Simplify

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

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

x \to 1 lim (\frac{x + 3}{x - 2})

6, 0, 16 - 2 x

\frac{d y}{d x} = \frac{x y}{3}

\frac{d}{d x} (x^{- 2} \cdot ln (x))

\int_{0}^{1} t^{2} e^{t} d t

What is (\partial)/(\partial y)((xy^2)/(x^2y^3+1)) ?
The answer to (\partial)/(\partial y)((xy^2)/(x^2y^3+1)) is (x(-x^2y^4+2y))/((x^2y^3+1)^2)