derivative (1+w)sqrt(w^3+9)

解

導関数

(1 + w) w^{3} + 9

\frac{3 w ^{2} + 5 w ^{3} + 18}{2 w ^{3} + 9}

解答ステップ

\frac{d}{d w} ((1 + w) w^{3} + 9)

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = 1 + w, g = w^{3} + 9

= \frac{d}{d w} (1 + w) w^{3} + 9 + \frac{d}{d w} (w^{3} + 9) (1 + w)

\frac{d}{d w} (1 + w) = 1

\frac{d}{d w} (w^{3} + 9) = \frac{3 w ^{2}}{2 w ^{3} + 9}

= 1 \cdot w^{3} + 9 + \frac{3 w ^{2}}{2 w ^{3} + 9} (1 + w)

簡素化

1 \cdot w^{3} + 9 + \frac{3 w ^{2}}{2 w ^{3} + 9} (1 + w) : \frac{3 w ^{2} + 5 w ^{3} + 18}{2 w ^{3} + 9}

= \frac{3 w ^{2} + 5 w ^{3} + 18}{2 w ^{3} + 9}