What is (\partial)/(\partial x)((8x)/(x^2+6y^2)) ?

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

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(\partial)/(\partial x)((8x)/(x^2+6y^2))

\frac{\partial}{\partial x} (\frac{8 x}{x ^{2} + 6 y ^{2}})

\frac{8 ( - x ^{2} + 6 y ^{2} )}{( x ^{2} + 6 y ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{8 x}{x ^{2} + 6 y ^{2}})

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 8 \frac{\partial}{\partial x} (\frac{x}{x ^{2} + 6 y ^{2}})

Apply the Quotient Rule:

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

= 8 \cdot \frac{\frac{\partial}{\partial x} ( x ) ( x ^{2} + 6 y ^{2} ) - \frac{\partial}{\partial x} ( x ^{2} + 6 y ^{2} ) x}{( x ^{2} + 6 y ^{2} ) ^{2}}

\frac{\partial}{\partial x} (x) = 1

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

= 8 \cdot \frac{1 \cdot ( x ^{2} + 6 y ^{2} ) - 2 xx}{( x ^{2} + 6 y ^{2} ) ^{2}}

Simplify

8 \cdot \frac{1 \cdot ( x ^{2} + 6 y ^{2} ) - 2 xx}{( x ^{2} + 6 y ^{2} ) ^{2}} : \frac{8 ( - x ^{2} + 6 y ^{2} )}{( x ^{2} + 6 y ^{2} ) ^{2}}

= \frac{8 ( - x ^{2} + 6 y ^{2} )}{( x ^{2} + 6 y ^{2} ) ^{2}}

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