d/(dz)(ln(sqrt((c^2-z^2)/(c^2+z^2))))

解

\frac{d}{d z} (ln (\frac{c ^{2} - z ^{2}}{c ^{2} + z ^{2}}))

- \frac{2 c ^{2} z}{( z ^{2} + c ^{2} ) ( - z ^{2} + c ^{2} )}

解答ステップ

\frac{d}{d z} (ln (\frac{c ^{2} - z ^{2}}{c ^{2} + z ^{2}}))

連鎖法則を適用:

\frac{1}{\frac{c ^{2} - z ^{2}}{c ^{2} + z ^{2}}} \frac{d}{d z} (\frac{c ^{2} - z ^{2}}{c ^{2} + z ^{2}})

= \frac{1}{\frac{c ^{2} - z ^{2}}{c ^{2} + z ^{2}}} \frac{d}{d z} (\frac{c ^{2} - z ^{2}}{c ^{2} + z ^{2}})

\frac{d}{d z} (\frac{c ^{2} - z ^{2}}{c ^{2} + z ^{2}}) = - \frac{2 c ^{2} z}{- z ^{2} + c ^{2} ( z ^{2} + c ^{2} ) ^{\frac{3}{2}}}

= \frac{1}{\frac{c ^{2} - z ^{2}}{c ^{2} + z ^{2}}} (- \frac{2 c ^{2} z}{- z ^{2} + c ^{2} ( z ^{2} + c ^{2} ) ^{\frac{3}{2}}})

簡素化

\frac{1}{\frac{c ^{2} - z ^{2}}{c ^{2} + z ^{2}}} (- \frac{2 c ^{2} z}{- z ^{2} + c ^{2} ( z ^{2} + c ^{2} ) ^{\frac{3}{2}}}) : - \frac{2 c ^{2} z}{( z ^{2} + c ^{2} ) ( - z ^{2} + c ^{2} )}

= - \frac{2 c ^{2} z}{( z ^{2} + c ^{2} ) ( - z ^{2} + c ^{2} )}