derivative ln((6x^2+5)/(x^{4/5)})

解

導関数

ln (\frac{6 x ^{2} + 5}{x ^{\frac{4}{5}}})

\frac{36 x ^{2} - 20}{5 x ( 6 x ^{2} + 5 )}

解答ステップ

\frac{d}{d x} (ln (\frac{6 x ^{2} + 5}{x ^{\frac{4}{5}}}))

連鎖法則を適用:

\frac{1}{\frac{6 x ^{2} + 5}{x ^{\frac{4}{5}}}} \frac{d}{d x} (\frac{6 x ^{2} + 5}{x ^{\frac{4}{5}}})

= \frac{1}{\frac{6 x ^{2} + 5}{x ^{\frac{4}{5}}}} \frac{d}{d x} (\frac{6 x ^{2} + 5}{x ^{\frac{4}{5}}})

\frac{d}{d x} (\frac{6 x ^{2} + 5}{x ^{\frac{4}{5}}}) = \frac{36 x ^{2} - 20}{5 x ^{\frac{9}{5}}}

= \frac{1}{\frac{6 x ^{2} + 5}{x ^{\frac{4}{5}}}} \cdot \frac{36 x ^{2} - 20}{5 x ^{\frac{9}{5}}}

簡素化

\frac{1}{\frac{6 x ^{2} + 5}{x ^{\frac{4}{5}}}} \cdot \frac{36 x ^{2} - 20}{5 x ^{\frac{9}{5}}} : \frac{36 x ^{2} - 20}{5 x ( 6 x ^{2} + 5 )}

= \frac{36 x ^{2} - 20}{5 x ( 6 x ^{2} + 5 )}