derivative of (3x^2+1/(2x))

解

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

\frac{3 x ^{2} - 1}{2 x ^{2}}

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

\frac{d}{d x} (3 x^{2} + 1) = 6 x

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

= \frac{1}{2} \cdot \frac{6 xx - 1 \cdot ( 3 x ^{2} + 1 )}{x ^{2}}

簡素化

\frac{1}{2} \cdot \frac{6 xx - 1 \cdot ( 3 x ^{2} + 1 )}{x ^{2}} : \frac{3 x ^{2} - 1}{2 x ^{2}}

= \frac{3 x ^{2} - 1}{2 x ^{2}}