d/(dt)(e^t(sin(t)+cos(t)))

解

\frac{d}{d t} (e^{t} (sin (t) + cos (t)))

2 e^{t} cos (t)

解答ステップ

\frac{d}{d t} (e^{t} (sin (t) + cos (t)))

積の法則を適用:

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

f = e^{t}, g = sin (t) + cos (t)

= \frac{d}{d t} (e^{t}) (sin (t) + cos (t)) + \frac{d}{d t} (sin (t) + cos (t)) e^{t}

\frac{d}{d t} (e^{t}) = e^{t}

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

= e^{t} (sin (t) + cos (t)) + (cos (t) - sin (t)) e^{t}

簡素化

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

= 2 e^{t} cos (t)