derivative of ln((x-1^3(x^2+1)^4))

解

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

\frac{11 x ^{2} - 8 x + 3}{( x - 1 ) ( x ^{2} + 1 )}

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

\frac{1}{( x - 1 ) ^{3} ( x ^{2} + 1 ) ^{4}} (3 (x - 1)^{2} (x^{2} + 1)^{4} + 8 x (x^{2} + 1)^{3} (x - 1)^{3}) : \frac{11 x ^{2} - 8 x + 3}{( x - 1 ) ( x ^{2} + 1 )}

= \frac{11 x ^{2} - 8 x + 3}{( x - 1 ) ( x ^{2} + 1 )}