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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Simplify

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

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

x^{3} - 11 x^{2} + 18 x, - x^{3} + 11 x^{2} - 18 x

\int \frac{- ln ( x )}{2} d x

f (x) = x^{2} 1 - x^{2}

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

f (x) = \frac{cos ( x ) + 4}{1 - sin ( x )}

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