Calculus and Its Applications, Chapter 1, 1.7, Section 1.7, Problem 32

Differentiate $\displaystyle f(x) = \left( \frac{2x}{x^2 + 1} \right)^3 $.



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\begin{equation}
\begin{aligned}

f'(x) =& 3 \left( \frac{2x}{x^2 + 1} \right)^2 \cdot \frac{d}{dx} \left( \frac{2x}{x^2 + 1} \right)
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f'(x) =& 3 \left( \frac{2x}{x^2 + 1} \right)^2 \left[ \frac{ \displaystyle (x^2 + 1) \cdot \frac{d}{dx} (2x) - 2x \cdot \frac{d}{dx} (x^2 + 1) }{(x^2 + 1)^2} \right]
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f'(x) =& 3 \left( \frac{2x}{x^2 + 1} \right)^2 \left[ \frac{(x^2 + 1)(2) - 2x (2x)}{(x^2 + 1)^2} \right]
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f'(x) =& 3 \left( \frac{2x}{x^2 + 1} \right)^2 \left[ \frac{2x^2 +2 - 4x^2}{(x^2 + 1)^2} \right]
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f'(x) =& 3 \left( \frac{2x}{x^2 + 1} \right)^2 \left[ \frac{2-2x^2}{(x^2 + 1)^2} \right]
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f'(x) =& \frac{3(2x)^2 (2-2x^2)}{(x^2 + 1)^2 (x^2 + 1)^2}
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f'(x) =& \frac{12x^2 (2-2x^2)}{(x^2 + 1)^4}

\end{aligned}
\end{equation}
$