What is (\partial)/(\partial x)((-5x-y)/(y^2+x^2)) ?

The answer to (\partial)/(\partial x)((-5x-y)/(y^2+x^2)) is (-5y^2+5x^2+2xy)/((y^2+x^2)^2)

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(\partial)/(\partial x)((-5x-y)/(y^2+x^2))

\frac{\partial}{\partial x} (\frac{- 5 x - y}{y ^{2} + x ^{2}})

\frac{- 5 y ^{2} + 5 x ^{2} + 2 x y}{( y ^{2} + x ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{- 5 x - y}{y ^{2} + x ^{2}})

Treat

y

as a constant

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{\partial}{\partial x} ( - 5 x - y ) ( y ^{2} + x ^{2} ) - \frac{\partial}{\partial x} ( y ^{2} + x ^{2} ) ( - 5 x - y )}{( y ^{2} + x ^{2} ) ^{2}}

\frac{\partial}{\partial x} (- 5 x - y) = - 5

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

= \frac{( - 5 ) ( y ^{2} + x ^{2} ) - 2 x ( - 5 x - y )}{( y ^{2} + x ^{2} ) ^{2}}

Simplify

\frac{( - 5 ) ( y ^{2} + x ^{2} ) - 2 x ( - 5 x - y )}{( y ^{2} + x ^{2} ) ^{2}} : \frac{- 5 y ^{2} + 5 x ^{2} + 2 x y}{( y ^{2} + x ^{2} ) ^{2}}

= \frac{- 5 y ^{2} + 5 x ^{2} + 2 x y}{( y ^{2} + x ^{2} ) ^{2}}

What is (\partial)/(\partial x)((-5x-y)/(y^2+x^2)) ?
The answer to (\partial)/(\partial x)((-5x-y)/(y^2+x^2)) is (-5y^2+5x^2+2xy)/((y^2+x^2)^2)