derivative of arctanh(\sqrt[4]{x})

解

\frac{d}{d x} (arctanh (4 x))

\frac{1}{4 x ^{\frac{3}{4}} ( 1 - x )}

解答ステップ

\frac{d}{d x} (arctanh (4 x))

連鎖法則を適用:

\frac{1}{1 - ( 4 x ) ^{2}} \frac{d}{d x} (4 x)

= \frac{1}{1 - ( 4 x ) ^{2}} \frac{d}{d x} (4 x)

\frac{d}{d x} (4 x) = \frac{1}{4 x ^{\frac{3}{4}}}

= \frac{1}{1 - ( 4 x ) ^{2}} \cdot \frac{1}{4 x ^{\frac{3}{4}}}

簡素化

\frac{1}{1 - ( 4 x ) ^{2}} \cdot \frac{1}{4 x ^{\frac{3}{4}}} : \frac{1}{4 x ^{\frac{3}{4}} ( 1 - x )}

= \frac{1}{4 x ^{\frac{3}{4}} ( 1 - x )}