derivative of (t(e^t(sin(t)+cos(t))))

解

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

e^{x} (sin (x) + cos (x)) + 2 e^{x} x cos (x)

解答ステップ

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

積の法則を適用:

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

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

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

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

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

= 1 \cdot e^{x} (sin (x) + cos (x)) + 2 e^{x} cos (x) x

簡素化

= e^{x} (sin (x) + cos (x)) + 2 e^{x} x cos (x)