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

解

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

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

解答ステップ

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

d

を定数として扱う

連鎖法則を適用:

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

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

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

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

簡素化

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

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