derivative 1/(sqrt(64+5t^2))

解

導関数

\frac{1}{64 + 5 t ^{2}}

- \frac{5 t}{( 64 + 5 t ^{2} ) ^{\frac{3}{2}}}

解答ステップ

\frac{d}{d t} (\frac{1}{64 + 5 t ^{2}})

指数の規則を適用する:

\frac{1}{a} = a^{- 1}

= \frac{d}{d t} ((64 + 5 t^{2})^{- \frac{1}{2}})

連鎖法則を適用:

- \frac{1}{2 ( 64 + 5 t ^{2} ) ^{\frac{3}{2}}} \frac{d}{d t} (64 + 5 t^{2})

= - \frac{1}{2 ( 64 + 5 t ^{2} ) ^{\frac{3}{2}}} \frac{d}{d t} (64 + 5 t^{2})

\frac{d}{d t} (64 + 5 t^{2}) = 10 t

= - \frac{1}{2 ( 64 + 5 t ^{2} ) ^{\frac{3}{2}}} \cdot 10 t

簡素化

- \frac{1}{2 ( 64 + 5 t ^{2} ) ^{\frac{3}{2}}} \cdot 10 t : - \frac{5 t}{( 64 + 5 t ^{2} ) ^{\frac{3}{2}}}

= - \frac{5 t}{( 64 + 5 t ^{2} ) ^{\frac{3}{2}}}