derivative f(x)=-(xsin((x^2)/4))/(2)

解

導関数

f (x) = - \frac{x sin ( \frac{x ^{2}}{4} )}{2}

- \frac{1}{2} sin (\frac{x ^{2}}{4}) + \frac{x ^{2} cos ( \frac{x ^{2}}{4} )}{2}

解答ステップ

\frac{d}{d x} - \frac{x sin ( \frac{x ^{2}}{4} )}{2}

定数を除去:

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

= - \frac{1}{2} \frac{d}{d x} (x sin (\frac{x ^{2}}{4}))

積の法則を適用:

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

f = x, g = sin (\frac{x ^{2}}{4})

= - \frac{1}{2} (\frac{d x}{d x} sin (\frac{x ^{2}}{4}) + \frac{d}{d x} (sin (\frac{x ^{2}}{4})) x)

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

\frac{d}{d x} (sin (\frac{x ^{2}}{4})) = \frac{x cos ( \frac{x ^{2}}{4} )}{2}

= - \frac{1}{2} 1 \cdot sin (\frac{x ^{2}}{4}) + \frac{x cos ( \frac{x ^{2}}{4} )}{2} x

簡素化

- \frac{1}{2} 1 \cdot sin (\frac{x ^{2}}{4}) + \frac{x cos ( \frac{x ^{2}}{4} )}{2} x : - \frac{1}{2} sin (\frac{x ^{2}}{4}) + \frac{x ^{2} cos ( \frac{x ^{2}}{4} )}{2}

= - \frac{1}{2} sin (\frac{x ^{2}}{4}) + \frac{x ^{2} cos ( \frac{x ^{2}}{4} )}{2}