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

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

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

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

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

Solution steps

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

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

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

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

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

Simplify

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

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

\frac{\partial}{\partial x} (4 - x^{2} - 4 Y (x)^{2})

\frac{d}{d x} (- x^{3} + 5 x^{2} - 6 x)

\int \frac{( x ^{2} )}{1 + x ^{2}} d x

y = \frac{4}{5 - 3 ( x )}

x \to 4 lim (\frac{x}{x - 4})

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