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

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

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

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

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

Solution steps

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

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

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

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

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

Expand

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

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

\int \frac{sin ( a x )}{1 - cos ( a x )} d x

x \to 0 lim (\frac{5}{x ^{2} - x})

\frac{d}{d x} (\frac{32 x}{( x ^{2} + 3 ) ^{2}})

4 t^{- \frac{3}{8}}

\int \frac{( x + 7 ) ^{7}}{2 x} d x

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