Single Variable Calculus, Chapter 7, 7.2-2, Section 7.2-2, Problem 26

Differentiate $\displaystyle H(x) = \ln \sqrt{\frac{a^2 - x^2}{a^2 + x^2}} $


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\begin{equation}
\begin{aligned}

\text{if } H(x) =& \ln \sqrt{\frac{a^2 - x^2}{a^2 + x^2}}, \text{ then by applying Chain Rule and Quotient Rule}
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H'(x) =& \frac{\displaystyle \frac{1}{2} \left( \frac{a^2 - x^2}{a^2 + x^2} \right)^{\frac{1}{2} - 1} \cdot \left[ \frac{(a^2 + x^2)(-2x) - (a^2 - x^2)(2x) }{(a^2 + x^2)^2} \right] }{\displaystyle \left( \frac{a^2 - x^2}{a^2 + x^2} \right)^{\frac{1}{2}}}
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H'(x) =& \frac{\displaystyle \frac{-2a^2x - 2x^3 - 2a^2x + 2x^3}{(a^2 + x^2)^2}}{2 \left( \frac{a^2 - x^2}{a^2 + x^2} \right)}
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H'(x) =& \frac{-4a^2 x}{2 (a^2 + x^2)(a^2 - x^2)}
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H'(x) =& \frac{-2a^2x}{(a^2 + x^2)(a^2 - x^2)}
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H'(x) =& \frac{-2a^2x}{a^4 - x^4}

\end{aligned}
\end{equation}
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