derivative of 12x^3cos(x^4)

解

\frac{d}{d x} (12 x^{3} cos (x^{4}))

12 (3 x^{2} cos (x^{4}) - 4 x^{6} sin (x^{4}))

解答ステップ

\frac{d}{d x} (12 x^{3} cos (x^{4}))

定数を除去:

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

= 12 \frac{d}{d x} (x^{3} cos (x^{4}))

積の法則を適用:

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

f = x^{3}, g = cos (x^{4})

= 12 (\frac{d}{d x} (x^{3}) cos (x^{4}) + \frac{d}{d x} (cos (x^{4})) x^{3})

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

\frac{d}{d x} (cos (x^{4})) = - sin (x^{4}) \cdot 4 x^{3}

= 12 (3 x^{2} cos (x^{4}) + (- sin (x^{4}) \cdot 4 x^{3}) x^{3})

簡素化

12 (3 x^{2} cos (x^{4}) + (- sin (x^{4}) \cdot 4 x^{3}) x^{3}) : 12 (3 x^{2} cos (x^{4}) - 4 x^{6} sin (x^{4}))

= 12 (3 x^{2} cos (x^{4}) - 4 x^{6} sin (x^{4}))