d/(dt)(cos(t)(1-sin(t)))

解

\frac{d}{d t} (cos (t) (1 - sin (t)))

- sin (t) (1 - sin (t)) - cos^{2} (t)

解答ステップ

\frac{d}{d t} (cos (t) (1 - sin (t)))

積の法則を適用:

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

f = cos (t), g = 1 - sin (t)

= \frac{d}{d t} (cos (t)) (1 - sin (t)) + \frac{d}{d t} (1 - sin (t)) cos (t)

\frac{d}{d t} (cos (t)) = - sin (t)

\frac{d}{d t} (1 - sin (t)) = - cos (t)

= (- sin (t)) (1 - sin (t)) + (- cos (t)) cos (t)

簡素化

(- sin (t)) (1 - sin (t)) + (- cos (t)) cos (t) : - sin (t) (1 - sin (t)) - cos^{2} (t)

= - sin (t) (1 - sin (t)) - cos^{2} (t)