derivative (t^2+1)(t-1/t)

解

導関数

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

3 t^{2} + \frac{1}{t ^{2}}

解答ステップ

\frac{d}{d t} ((t^{2} + 1) (t - \frac{1}{t}))

積の法則を適用:

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

f = t^{2} + 1, g = t - \frac{1}{t}

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

\frac{d}{d t} (t^{2} + 1) = 2 t

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

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

簡素化

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

= 3 t^{2} + \frac{1}{t ^{2}}