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

The answer to (\partial)/(\partial x)((3y)/(x^2+y^2)) is -(6yx)/((x^2+y^2)^2)

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

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

- \frac{6 y x}{( x ^{2} + y ^{2} ) ^{2}}

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

= 3 y \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

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

Simplify

3 y (- \frac{1}{( x ^{2} + y ^{2} ) ^{2}} \cdot 2 x) : - \frac{6 y x}{( x ^{2} + y ^{2} ) ^{2}}

= - \frac{6 y x}{( x ^{2} + y ^{2} ) ^{2}}

\int (9 x (x^{8} - \frac{1}{9})) d x

\frac{d}{d x} (\frac{tan ( x ) - 4}{sec ( x )})

\frac{d}{d x} (\frac{7}{( x ^{2} - 1 ) ^{3}})

\frac{d}{d x} (- 2 x^{3} - 3 x^{2} + 12 x + 11)

f (x) = 3 x^{2} + 5 x, at x = - 3

What is (\partial)/(\partial x)((3y)/(x^2+y^2)) ?
The answer to (\partial)/(\partial x)((3y)/(x^2+y^2)) is -(6yx)/((x^2+y^2)^2)