derivative of ln((x^3-2x/(4+x)))

解

\frac{d}{d x} (ln (\frac{x ^{3} - 2 x}{4 + x}))

\frac{2 x ^{3} + 12 x ^{2} - 8}{( x ^{3} - 2 x ) ( x + 4 )}

解答ステップ

\frac{d}{d x} (ln (\frac{x ^{3} - 2 x}{4 + x}))

連鎖法則を適用:

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

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

\frac{d}{d x} (\frac{x ^{3} - 2 x}{4 + x}) = \frac{2 x ^{3} + 12 x ^{2} - 8}{( 4 + x ) ^{2}}

= \frac{1}{\frac{x ^{3} - 2 x}{4 + x}} \cdot \frac{2 x ^{3} + 12 x ^{2} - 8}{( 4 + x ) ^{2}}

簡素化

\frac{1}{\frac{x ^{3} - 2 x}{4 + x}} \cdot \frac{2 x ^{3} + 12 x ^{2} - 8}{( 4 + x ) ^{2}} : \frac{2 x ^{3} + 12 x ^{2} - 8}{( x ^{3} - 2 x ) ( x + 4 )}

= \frac{2 x ^{3} + 12 x ^{2} - 8}{( x ^{3} - 2 x ) ( x + 4 )}