Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 6 - Inverse Functions - 6.4 Derivatives of Logarithmic Functions - 6.4 Exercises: 22

Answer

$-\frac{2a^2z}{a^4-z^4}$

Work Step by Step

$\frac{d}{dz}\ln\sqrt{\frac{a^2-z^2}{a^2+z^2}}$ In this case, it is much easier to simplify the expression (using the logarithm laws on page 439) before taking the derivative. $=\frac{d}{dz}\frac{1}{2}\ln\frac{a^2-z^2}{a^2+z^2}$ $=\frac{d}{dz}\frac{1}{2}(\ln(a^2-z^2)-\ln(a^2+z^2))$ $=\frac{1}{2}(\frac{1}{a^2-z^2}\frac{d}{dz}(a^2-z^2)-\frac{1}{a^2+z^2}\frac{d}{dz}(a^2+z^2))$ $=\frac{1}{2}(\frac{-2z}{a^2-z^2}-\frac{2z}{a^2+z^2})$ $=-\frac{z}{a^2-z^2}-\frac{z}{a^2+z^2}$ $=-\frac{z}{a^2-z^2}*\frac{a^2+z^2}{a^2+z^2}-\frac{z}{a^2+z^2}*\frac{a^2-z^2}{a^2-z^2}$ $=-\frac{a^2z+z^3}{a^4-z^4}-\frac{a^2z-z^3}{a^4-z^4}$ $=\frac{-(a^2z+z^3)-(a^2z-z^3)}{a^4-z^4}$ $=\frac{-a^2z-z^3-a^2z+z^3}{a^4-z^4}$ $=-\frac{2a^2z}{a^4-z^4}$
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