What is (\partial)/(\partial x)((7y-4x)/(y^2+2x^2)) ?

The answer to (\partial)/(\partial x)((7y-4x)/(y^2+2x^2)) is (8x^2-28yx-4y^2)/((y^2+2x^2)^2)

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(\partial)/(\partial x)((7y-4x)/(y^2+2x^2))

\frac{\partial}{\partial x} (\frac{7 y - 4 x}{y ^{2} + 2 x ^{2}})

\frac{8 x ^{2} - 28 y x - 4 y ^{2}}{( y ^{2} + 2 x ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{7 y - 4 x}{y ^{2} + 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} ( 7 y - 4 x ) ( y ^{2} + 2 x ^{2} ) - \frac{\partial}{\partial x} ( y ^{2} + 2 x ^{2} ) ( 7 y - 4 x )}{( y ^{2} + 2 x ^{2} ) ^{2}}

\frac{\partial}{\partial x} (7 y - 4 x) = - 4

\frac{\partial}{\partial x} (y^{2} + 2 x^{2}) = 4 x

= \frac{( - 4 ) ( y ^{2} + 2 x ^{2} ) - 4 x ( 7 y - 4 x )}{( y ^{2} + 2 x ^{2} ) ^{2}}

Simplify

\frac{( - 4 ) ( y ^{2} + 2 x ^{2} ) - 4 x ( 7 y - 4 x )}{( y ^{2} + 2 x ^{2} ) ^{2}} : \frac{8 x ^{2} - 28 y x - 4 y ^{2}}{( y ^{2} + 2 x ^{2} ) ^{2}}

= \frac{8 x ^{2} - 28 y x - 4 y ^{2}}{( y ^{2} + 2 x ^{2} ) ^{2}}

\frac{d}{d x} (30 x)

\int (x^{8} - 4 x^{2})^{3} (x^{7} - x) d x

\frac{\partial}{\partial z} (x e^{y})

y = \frac{1}{2 x + 7}

f (x) = 2 x + \frac{5}{x}

What is (\partial)/(\partial x)((7y-4x)/(y^2+2x^2)) ?
The answer to (\partial)/(\partial x)((7y-4x)/(y^2+2x^2)) is (8x^2-28yx-4y^2)/((y^2+2x^2)^2)