derivative of ((x^2/(x^2+2))^2)

解答

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

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

求解步骤

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

使用连锁律:

2 \cdot \frac{x ^{2}}{x ^{2} + 2} \frac{d}{d x} (\frac{x ^{2}}{x ^{2} + 2})

= 2 \cdot \frac{x ^{2}}{x ^{2} + 2} \frac{d}{d x} (\frac{x ^{2}}{x ^{2} + 2})

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

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

化简

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

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

\frac{d ^{2}}{d x ^{2}} (4 x - 5 x^{\frac{3}{4}})

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