derivative of ((1+3x/(3x))(3-x))

解

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

- \frac{x ^{2} + 1}{x ^{2}}

解答ステップ

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

定数を除去:

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

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

積の法則を適用:

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

f = \frac{1 + 3 x}{x}, g = 3 - x

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

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

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

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

簡素化

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

= - \frac{x ^{2} + 1}{x ^{2}}