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

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


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