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

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

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

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

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

Solution steps

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

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

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

= \frac{1 \cdot ( 2 x ^{2} + y ^{2} ) - 4 xx}{( 2 x ^{2} + y ^{2} ) ^{2}}

1 \cdot (2 x^{2} + y^{2}) - 4 xx = - 2 x^{2} + y^{2}

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

\int x^{3} \cdot e^{12 x} d x

x^{3} + 3 x^{2}

e^{s i n (2 x)} + sin (e^{2 x})

\frac{1}{1 + x ^{3}}, 0

\frac{d}{d x} (2 e^{4 x^{2}})

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