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

The answer to (\partial)/(\partial x)(((x^2+2)/(x^2-2))^3) is -(24x(x^2+2)^2)/((x^2-2)^4)

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(\partial)/(\partial x)(((x^2+2)/(x^2-2))^3)

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

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

Solution steps

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

Apply the chain rule:

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

= 3 (\frac{x ^{2} + 2}{x ^{2} - 2})^{2} \frac{\partial}{\partial x} (\frac{x ^{2} + 2}{x ^{2} - 2})

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

= 3 (\frac{x ^{2} + 2}{x ^{2} - 2})^{2} (- \frac{8 x}{( x ^{2} - 2 ) ^{2}})

Simplify

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

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

\int \frac{1}{( 81 + x ^{2} ) ^{\frac{3}{2}}} d x

x \to 0 lim (1 - cos (4 x))

1000 y^{^{''}} - 3 y^{^{'}} + 17 y = 0

\frac{d ^{6}}{d x ^{6}} ((2 x + 5)^{7})

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

What is (\partial)/(\partial x)(((x^2+2)/(x^2-2))^3) ?
The answer to (\partial)/(\partial x)(((x^2+2)/(x^2-2))^3) is -(24x(x^2+2)^2)/((x^2-2)^4)