derivative of ln((1+sqrt(x)/(x^4)))

解

\frac{d}{d x} (ln (\frac{1 + x}{x ^{4}}))

\frac{x ^{\frac{7}{2}} - 8 x ^{3} ( x + 1 )}{2 x ^{4} ( x + 1 )}

解答ステップ

\frac{d}{d x} (ln (\frac{1 + x}{x ^{4}}))

連鎖法則を適用:

\frac{1}{\frac{1 + x}{x ^{4}}} \frac{d}{d x} (\frac{1 + x}{x ^{4}})

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

\frac{d}{d x} (\frac{1 + x}{x ^{4}}) = \frac{x ^{\frac{7}{2}} - 8 x ^{3} ( 1 + x )}{2 x ^{8}}

= \frac{1}{\frac{1 + x}{x ^{4}}} \cdot \frac{x ^{\frac{7}{2}} - 8 x ^{3} ( 1 + x )}{2 x ^{8}}

簡素化

\frac{1}{\frac{1 + x}{x ^{4}}} \cdot \frac{x ^{\frac{7}{2}} - 8 x ^{3} ( 1 + x )}{2 x ^{8}} : \frac{x ^{\frac{7}{2}} - 8 x ^{3} ( x + 1 )}{2 x ^{4} ( x + 1 )}

= \frac{x ^{\frac{7}{2}} - 8 x ^{3} ( x + 1 )}{2 x ^{4} ( x + 1 )}