What is (\partial)/(\partial y)((-x-7y)/(y^2+2x^2)) ?

The answer to (\partial)/(\partial y)((-x-7y)/(y^2+2x^2)) is (7y^2+2xy-14x^2)/((y^2+2x^2)^2)

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(\partial)/(\partial y)((-x-7y)/(y^2+2x^2))

\frac{\partial}{\partial y} (\frac{- x - 7 y}{y ^{2} + 2 x ^{2}})

\frac{7 y ^{2} + 2 x y - 14 x ^{2}}{( y ^{2} + 2 x ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial y} (\frac{- x - 7 y}{y ^{2} + 2 x ^{2}})

Treat

x

as a constant

Apply the Quotient Rule:

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

= \frac{\frac{\partial}{\partial y} ( - x - 7 y ) ( y ^{2} + 2 x ^{2} ) - \frac{\partial}{\partial y} ( y ^{2} + 2 x ^{2} ) ( - x - 7 y )}{( y ^{2} + 2 x ^{2} ) ^{2}}

\frac{\partial}{\partial y} (- x - 7 y) = - 7

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

= \frac{( - 7 ) ( y ^{2} + 2 x ^{2} ) - 2 y ( - x - 7 y )}{( y ^{2} + 2 x ^{2} ) ^{2}}

Simplify

\frac{( - 7 ) ( y ^{2} + 2 x ^{2} ) - 2 y ( - x - 7 y )}{( y ^{2} + 2 x ^{2} ) ^{2}} : \frac{7 y ^{2} + 2 x y - 14 x ^{2}}{( y ^{2} + 2 x ^{2} ) ^{2}}

= \frac{7 y ^{2} + 2 x y - 14 x ^{2}}{( y ^{2} + 2 x ^{2} ) ^{2}}

\int \frac{1}{x ( 1 + x ) ^{2}} d x

x \to 4 lim (\frac{- x + 4}{x})

\frac{d}{d x} (e^{2 x} (x - 4) (x + 6))

\int sin^{8} (x) \cdot cos^{5} (x) d x

\frac{d}{d x} ((2 x + 8)^{2})

What is (\partial)/(\partial y)((-x-7y)/(y^2+2x^2)) ?
The answer to (\partial)/(\partial y)((-x-7y)/(y^2+2x^2)) is (7y^2+2xy-14x^2)/((y^2+2x^2)^2)