d/(dt)(ln(a+{v}(t)t))

解

\frac{d}{d t} (ln (a + v (t) t))

\frac{t \frac{d}{d t} ( v ( t ) ) + v ( t )}{a + t v ( t )}

解答ステップ

\frac{d}{d t} (ln (a + v (t) t))

連鎖法則を適用:

\frac{1}{a + v ( t ) t} \frac{d}{d t} (a + v (t) t)

= \frac{1}{a + v ( t ) t} \frac{d}{d t} (a + v (t) t)

\frac{d}{d t} (a + v (t) t) = t \frac{d}{d t} (v (t)) + v (t)

= \frac{1}{a + v ( t ) t} (t \frac{d}{d t} (v (t)) + v (t))

簡素化

\frac{1}{a + v ( t ) t} (t \frac{d}{d t} (v (t)) + v (t)) : \frac{t \frac{d}{d t} ( v ( t ) ) + v ( t )}{a + t v ( t )}

= \frac{t \frac{d}{d t} ( v ( t ) ) + v ( t )}{a + t v ( t )}