derivative of sqrt((ln(x+1)/(ln(x-1))))

解

\frac{d}{d x} (\frac{ln ( x + 1 )}{ln ( x - 1 )})

\frac{ln ( x - 1 ) ( x - 1 ) - ln ( x + 1 ) ( x + 1 )}{2 ln ( x + 1 ) ln ^{\frac{3}{2}} ( x - 1 ) ( x + 1 ) ( x - 1 )}

解答ステップ

\frac{d}{d x} (\frac{ln ( x + 1 )}{ln ( x - 1 )})

連鎖法則を適用:

\frac{1}{2 \frac{l n ( x + 1 )}{l n ( x - 1 )}} \frac{d}{d x} (\frac{ln ( x + 1 )}{ln ( x - 1 )})

= \frac{1}{2 \frac{l n ( x + 1 )}{l n ( x - 1 )}} \frac{d}{d x} (\frac{ln ( x + 1 )}{ln ( x - 1 )})

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

= \frac{1}{2 \frac{l n ( x + 1 )}{l n ( x - 1 )}} \cdot \frac{ln ( x - 1 ) ( x - 1 ) - ln ( x + 1 ) ( x + 1 )}{ln ^{2} ( x - 1 ) ( x + 1 ) ( x - 1 )}

簡素化

\frac{1}{2 \frac{l n ( x + 1 )}{l n ( x - 1 )}} \cdot \frac{ln ( x - 1 ) ( x - 1 ) - ln ( x + 1 ) ( x + 1 )}{ln ^{2} ( x - 1 ) ( x + 1 ) ( x - 1 )} : \frac{ln ( x - 1 ) ( x - 1 ) - ln ( x + 1 ) ( x + 1 )}{2 ln ( x + 1 ) ln ^{\frac{3}{2}} ( x - 1 ) ( x + 1 ) ( x - 1 )}

= \frac{ln ( x - 1 ) ( x - 1 ) - ln ( x + 1 ) ( x + 1 )}{2 ln ( x + 1 ) ln ^{\frac{3}{2}} ( x - 1 ) ( x + 1 ) ( x - 1 )}