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

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

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

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

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

Solution steps

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

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

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

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

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

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

\int x^{2} sin (5 x) d x

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\frac{s}{( s + 1 ) ( s ^{2} + 1 )}

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