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

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

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

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

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

Solution steps

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

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

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

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

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

Expand

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

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

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