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

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

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

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

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

Solution steps

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

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

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

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

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

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

\frac{d y}{d x} = \frac{1}{x} (y^{2} + 0.2 y)

\int \frac{4 x ^{2} + 1}{4 x} d x

\int 3 x^{2} - 5 x + \frac{7}{x} + 2 e^{3} d x

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

x \to 2 + lim (x^{2} + 1)

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