derivative f(x)= x/(32)(sqrt(x)+\sqrt[3]{x})

解

導関数

f (x) = \frac{x}{32} (x + 3 x)

\frac{9 x + 2 x ^{\frac{1}{3}} + 6 3 x}{192}

解答ステップ

\frac{d}{d x} (\frac{x}{32} (x + 3 x))

定数を除去:

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

= \frac{1}{32} \frac{d}{d x} (x (x + 3 x))

積の法則を適用:

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

f = x, g = x + 3 x

= \frac{1}{32} (\frac{d x}{d x} (x + 3 x) + \frac{d}{d x} (x + 3 x) x)

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

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

= \frac{1}{32} (1 \cdot (x + 3 x) + (\frac{1}{2 x} + \frac{1}{3 x ^{\frac{2}{3}}}) x)

簡素化

\frac{1}{32} (1 \cdot (x + 3 x) + (\frac{1}{2 x} + \frac{1}{3 x ^{\frac{2}{3}}}) x) : \frac{9 x + 2 x ^{\frac{1}{3}} + 6 3 x}{192}

= \frac{9 x + 2 x ^{\frac{1}{3}} + 6 3 x}{192}