What is (\partial)/(\partial x)((7x-5y)/(y^2+2x^2)) ?

The answer to (\partial)/(\partial x)((7x-5y)/(y^2+2x^2)) is (7y^2-14x^2+20xy)/((y^2+2x^2)^2)

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(\partial)/(\partial x)((7x-5y)/(y^2+2x^2))

\frac{\partial}{\partial x} (\frac{7 x - 5 y}{y ^{2} + 2 x ^{2}})

\frac{7 y ^{2} - 14 x ^{2} + 20 x y}{( y ^{2} + 2 x ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{7 x - 5 y}{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 x - 5 y ) ( y ^{2} + 2 x ^{2} ) - \frac{\partial}{\partial x} ( y ^{2} + 2 x ^{2} ) ( 7 x - 5 y )}{( y ^{2} + 2 x ^{2} ) ^{2}}

\frac{\partial}{\partial x} (7 x - 5 y) = 7

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

= \frac{7 ( y ^{2} + 2 x ^{2} ) - 4 x ( 7 x - 5 y )}{( y ^{2} + 2 x ^{2} ) ^{2}}

Expand

7 (y^{2} + 2 x^{2}) - 4 x (7 x - 5 y) : 7 y^{2} - 14 x^{2} + 20 x y

= \frac{7 y ^{2} - 14 x ^{2} + 20 x y}{( y ^{2} + 2 x ^{2} ) ^{2}}

What is (\partial)/(\partial x)((7x-5y)/(y^2+2x^2)) ?
The answer to (\partial)/(\partial x)((7x-5y)/(y^2+2x^2)) is (7y^2-14x^2+20xy)/((y^2+2x^2)^2)