ddx[x2x3+1]=(x3+1)(x2)1−(x2)(x3+1)1(x3+1)2=(x3+1)(2x)−(x2)(3x2)(x3+1)2=2x4+2x−3x4(x3+1)2=2x−x4(x3+1)2

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$\frac{d}{d x} [\frac{x^{2}}{x^{3} + 1}] =$

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$\frac{2 x + x^{4}}{{(x^{3} + 1)}^{2}}$

$\frac{2 x + x^{4}}{x^{3} + 1}$

$\frac{2 x - x^{4}}{x^{3} + 1}$

$\frac{2 x - x^{4}}{{(x^{3} + 1)}^{2}}$

answer is C.

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Detailed Solution

$\frac{d}{d x} [\frac{x^{2}}{x^{3} + 1}] = \frac{(x^{3} + 1) {(x^{2})}^{1} - (x^{2}) {(x^{3} + 1)}^{1}}{{(x^{3} + 1)}^{2}}$

$= \frac{(x^{3} + 1) (2 x) - (x^{2}) (3 x^{2})}{{(x^{3} + 1)}^{2}}$

$= \frac{2 x^{4} + 2 x - 3 x^{4}}{{(x^{3} + 1)}^{2}}$

$= \frac{2 x - x^{4}}{{(x^{3} + 1)}^{2}}$

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