derivative ln(4x^3+5x^2+2x+5)

解

導関数

ln (4 x^{3} + 5 x^{2} + 2 x + 5)

\frac{12 x ^{2} + 10 x + 2}{4 x ^{3} + 5 x ^{2} + 2 x + 5}

解答ステップ

\frac{d}{d x} (ln (4 x^{3} + 5 x^{2} + 2 x + 5))

連鎖法則を適用:

\frac{1}{4 x ^{3} + 5 x ^{2} + 2 x + 5} \frac{d}{d x} (4 x^{3} + 5 x^{2} + 2 x + 5)

= \frac{1}{4 x ^{3} + 5 x ^{2} + 2 x + 5} \frac{d}{d x} (4 x^{3} + 5 x^{2} + 2 x + 5)

\frac{d}{d x} (4 x^{3} + 5 x^{2} + 2 x + 5) = 12 x^{2} + 10 x + 2

= \frac{1}{4 x ^{3} + 5 x ^{2} + 2 x + 5} (12 x^{2} + 10 x + 2)

簡素化

\frac{1}{4 x ^{3} + 5 x ^{2} + 2 x + 5} (12 x^{2} + 10 x + 2) : \frac{12 x ^{2} + 10 x + 2}{4 x ^{3} + 5 x ^{2} + 2 x + 5}

= \frac{12 x ^{2} + 10 x + 2}{4 x ^{3} + 5 x ^{2} + 2 x + 5}