What is (\partial)/(\partial y)((x^3)/(x^2+y^2)) ?

The answer to (\partial)/(\partial y)((x^3)/(x^2+y^2)) is -(2x^3y)/((x^2+y^2)^2)

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(\partial)/(\partial y)((x^3)/(x^2+y^2))

\frac{\partial}{\partial y} (\frac{x ^{3}}{x ^{2} + y ^{2}})

- \frac{2 x ^{3} y}{( x ^{2} + y ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial y} (\frac{x ^{3}}{x ^{2} + y ^{2}})

Treat

x

as a constant

Take the constant out:

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

= x^{3} \frac{\partial}{\partial y} (\frac{1}{x ^{2} + y ^{2}})

Apply exponent rule:

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

= x^{3} \frac{\partial}{\partial y} ((x^{2} + y^{2})^{- 1})

Apply the chain rule:

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

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

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

= x^{3} (- \frac{1}{( x ^{2} + y ^{2} ) ^{2}} \cdot 2 y)

Simplify

x^{3} (- \frac{1}{( x ^{2} + y ^{2} ) ^{2}} \cdot 2 y) : - \frac{2 x ^{3} y}{( x ^{2} + y ^{2} ) ^{2}}

= - \frac{2 x ^{3} y}{( x ^{2} + y ^{2} ) ^{2}}

n = 1 \sum \infty (- 5 x)^{n}

\int (x + 4) (2 x + 1) d x

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

\frac{d}{d x} (cos (θ) x)

\int 5 x d x

What is (\partial)/(\partial y)((x^3)/(x^2+y^2)) ?
The answer to (\partial)/(\partial y)((x^3)/(x^2+y^2)) is -(2x^3y)/((x^2+y^2)^2)