derivative (t+e^t)(6-sqrt(t))

解

導関数

(t + e^{t}) (6 - t)

\frac{- 3 t - 2 e ^{t} t + 12 e ^{t} t - e ^{t}}{2 t} + 6

解答ステップ

\frac{d}{d t} ((t + e^{t}) (6 - t))

積の法則を適用:

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

f = t + e^{t}, g = 6 - t

= \frac{d}{d t} (t + e^{t}) (6 - t) + \frac{d}{d t} (6 - t) (t + e^{t})

\frac{d}{d t} (t + e^{t}) = 1 + e^{t}

\frac{d}{d t} (6 - t) = - \frac{1}{2 t}

= (1 + e^{t}) (6 - t) + (- \frac{1}{2 t}) (t + e^{t})

簡素化

(1 + e^{t}) (6 - t) + (- \frac{1}{2 t}) (t + e^{t}) : \frac{- 3 t - 2 e ^{t} t + 12 e ^{t} t - e ^{t}}{2 t} + 6

= \frac{- 3 t - 2 e ^{t} t + 12 e ^{t} t - e ^{t}}{2 t} + 6