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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

x \frac{d y}{d x} + y = x^{3}

f (x) = c cos (t) + t^{2} sin (t)

\int_{3}^{4} (x + 7 x^{2}) d x

ln (3) \cdot e^{x}

\int x^{3} sin (d) x d x

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