(d^2)/(dt^2)((v)(av(t)-bv(t)ln(v)))

解

\frac{d ^{2}}{d t ^{2}} ((v (t)) (a v (t) - b v (t) ln (v (t))))

2 a \frac{d}{d t} (v (t))^{2} + 2 a v (t) \frac{d ^{2}}{d t ^{2}} (v (t)) - 3 b \frac{d}{d t} (v (t))^{2} - 2 b ln (v (t)) \frac{d}{d t} (v (t))^{2} - 2 b ln (v (t)) v (t) \frac{d ^{2}}{d t ^{2}} (v (t)) - b v (t) \frac{d ^{2}}{d t ^{2}} (v (t))

解答ステップ

\frac{d ^{2}}{d t ^{2}} ((v (t)) (a v (t) - b v (t) ln (v (t))))

\frac{d}{d t} (v (t) (a v (t) - b v (t) ln (v (t)))) = 2 a v (t) \frac{d}{d t} (v (t)) - 2 b ln (v (t)) v (t) \frac{d}{d t} (v (t)) - b v (t) \frac{d}{d t} (v (t))

= \frac{d}{d t} (2 a v (t) \frac{d}{d t} (v (t)) - 2 b ln (v (t)) v (t) \frac{d}{d t} (v (t)) - b v (t) \frac{d}{d t} (v (t)))

\frac{d}{d t} (2 a v (t) \frac{d}{d t} (v (t)) - 2 b ln (v (t)) v (t) \frac{d}{d t} (v (t)) - b v (t) \frac{d}{d t} (v (t))) = 2 a \frac{d}{d t} (v (t))^{2} + 2 a v (t) \frac{d ^{2}}{d t ^{2}} (v (t)) - 3 b \frac{d}{d t} (v (t))^{2} - 2 b ln (v (t)) \frac{d}{d t} (v (t))^{2} - 2 b ln (v (t)) v (t) \frac{d ^{2}}{d t ^{2}} (v (t)) - b v (t) \frac{d ^{2}}{d t ^{2}} (v (t))

= 2 a \frac{d}{d t} (v (t))^{2} + 2 a v (t) \frac{d ^{2}}{d t ^{2}} (v (t)) - 3 b \frac{d}{d t} (v (t))^{2} - 2 b ln (v (t)) \frac{d}{d t} (v (t))^{2} - 2 b ln (v (t)) v (t) \frac{d ^{2}}{d t ^{2}} (v (t)) - b v (t) \frac{d ^{2}}{d t ^{2}} (v (t))