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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

- \frac{1}{( y ^{2} + x ^{2} ) ^{2}} \frac{\partial}{\partial x} (y^{2} + x^{2})

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

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

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

Simplify

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

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

\frac{d}{d x} (\frac{2 x ^{3} - x ^{2} + 5 x}{x ^{2} - 25})

\int \frac{6 - y}{y - 1} d y

\int_{2}^{8} (10 - x) d x

f (x) = (2 x^{7} - 3 x^{5} + 2)^{22}

\frac{s + 3}{s ^{2} + 3 s + 2}

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