derivative of (3x^3+9x^2/x)

解

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

3 (2 x + 3)

解答ステップ

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

簡素化

\frac{3 x ^{3} + 9 x ^{2}}{x} : 3 x (x + 3)

= \frac{d}{d x} (3 x (x + 3))

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= 3 \frac{d}{d x} (x (x + 3))

積の法則を適用:

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

f = x, g = x + 3

= 3 (\frac{d x}{d x} (x + 3) + \frac{d}{d x} (x + 3) x)

\frac{d x}{d x} = 1

\frac{d}{d x} (x + 3) = 1

= 3 (1 \cdot (x + 3) + 1 \cdot x)

簡素化

3 (1 \cdot (x + 3) + 1 \cdot x) : 3 (2 x + 3)

= 3 (2 x + 3)