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

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

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

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

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

Solution steps

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

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

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

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

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

Expand

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

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

\int x^{4 n} d x

f (x) = \frac{2 x - 1}{x + 1}, at x = 1

3 x - cos (x)

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

n \to \infty lim (0. 5^{n})

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