What is (\partial)/(\partial x)((4x-6y)/(y^2+2x^2)) ?

The answer to (\partial)/(\partial x)((4x-6y)/(y^2+2x^2)) is (4y^2-8x^2+24xy)/((y^2+2x^2)^2)

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(\partial)/(\partial x)((4x-6y)/(y^2+2x^2))

\frac{\partial}{\partial x} (\frac{4 x - 6 y}{y ^{2} + 2 x ^{2}})

\frac{4 y ^{2} - 8 x ^{2} + 24 x y}{( y ^{2} + 2 x ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{4 x - 6 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} ( 4 x - 6 y ) ( y ^{2} + 2 x ^{2} ) - \frac{\partial}{\partial x} ( y ^{2} + 2 x ^{2} ) ( 4 x - 6 y )}{( y ^{2} + 2 x ^{2} ) ^{2}}

\frac{\partial}{\partial x} (4 x - 6 y) = 4

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

= \frac{4 ( y ^{2} + 2 x ^{2} ) - 4 x ( 4 x - 6 y )}{( y ^{2} + 2 x ^{2} ) ^{2}}

Expand

4 (y^{2} + 2 x^{2}) - 4 x (4 x - 6 y) : 4 y^{2} - 8 x^{2} + 24 x y

= \frac{4 y ^{2} - 8 x ^{2} + 24 x y}{( y ^{2} + 2 x ^{2} ) ^{2}}

\frac{d}{d x} (sin (ln (cos (x))))

\frac{d}{d x} (\frac{5 x ^{2} - 9 x + 13}{2 x + 4})

x \to 6 lim (\frac{6 + x}{x})

\int \frac{2 x ^{3} - 1}{x + x ^{4}} d x

f (x) = tan^{2} (x^{3})

What is (\partial)/(\partial x)((4x-6y)/(y^2+2x^2)) ?
The answer to (\partial)/(\partial x)((4x-6y)/(y^2+2x^2)) is (4y^2-8x^2+24xy)/((y^2+2x^2)^2)