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

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

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(\partial)/(\partial x)((x^2+2)/(x^2-2))

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

- \frac{8 x}{( x ^{2} - 2 ) ^{2}}

Solution steps

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

Apply the Quotient Rule:

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

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

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

\frac{\partial}{\partial x} (x^{2} - 2) = 2 x

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

Simplify

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

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

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\frac{d}{d x} (ln (2 x + 3 y))

(2 x^{2} y + 2 y + 5) d x + (2 x^{3} + 2 x) d y = 0

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