What is (\partial)/(\partial x)((3y-6x)/(y^2+7x^2)) ?

The answer to (\partial)/(\partial x)((3y-6x)/(y^2+7x^2)) is (42x^2-42yx-6y^2)/((y^2+7x^2)^2)

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(\partial)/(\partial x)((3y-6x)/(y^2+7x^2))

\frac{\partial}{\partial x} (\frac{3 y - 6 x}{y ^{2} + 7 x ^{2}})

\frac{42 x ^{2} - 42 y x - 6 y ^{2}}{( y ^{2} + 7 x ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{3 y - 6 x}{y ^{2} + 7 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} ( 3 y - 6 x ) ( y ^{2} + 7 x ^{2} ) - \frac{\partial}{\partial x} ( y ^{2} + 7 x ^{2} ) ( 3 y - 6 x )}{( y ^{2} + 7 x ^{2} ) ^{2}}

\frac{\partial}{\partial x} (3 y - 6 x) = - 6

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

= \frac{( - 6 ) ( y ^{2} + 7 x ^{2} ) - 14 x ( 3 y - 6 x )}{( y ^{2} + 7 x ^{2} ) ^{2}}

Simplify

\frac{( - 6 ) ( y ^{2} + 7 x ^{2} ) - 14 x ( 3 y - 6 x )}{( y ^{2} + 7 x ^{2} ) ^{2}} : \frac{42 x ^{2} - 42 y x - 6 y ^{2}}{( y ^{2} + 7 x ^{2} ) ^{2}}

= \frac{42 x ^{2} - 42 y x - 6 y ^{2}}{( y ^{2} + 7 x ^{2} ) ^{2}}

\int 1 - 100 x^{2} d x

\int_{0}^{\frac{π}{2}} 10 sin^{2} (3 x) d x

x \to 1 + lim (x ln (x))

\frac{d}{d x} (3 3 x^{2})

x \to 8 lim (1 + 3 x)

What is (\partial)/(\partial x)((3y-6x)/(y^2+7x^2)) ?
The answer to (\partial)/(\partial x)((3y-6x)/(y^2+7x^2)) is (42x^2-42yx-6y^2)/((y^2+7x^2)^2)