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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the Quotient Rule:

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

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

\frac{\partial}{\partial y} (y) = 1

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

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

1 \cdot (x^{2} y^{2} + 1) - 2 x^{2} yy = - y^{2} x^{2} + 1

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

x^{2} + x^{3}

\int e^{t - s} d s

\frac{d}{d x} (2 π^{2})

\int \frac{x ^{4}}{x + 1} d x

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