d/(dt)(t/(sqrt(t^2+16)))

解

\frac{d}{d t} (\frac{t}{t ^{2} + 16})

\frac{16}{( t ^{2} + 16 ) t ^{2} + 16}

解答ステップ

\frac{d}{d t} (\frac{t}{t ^{2} + 16})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d t}{d t} t ^{2} + 16 - \frac{d}{d t} ( t ^{2} + 16 ) t}{( t ^{2} + 16 ) ^{2}}

\frac{d t}{d t} = 1

\frac{d}{d t} (t^{2} + 16) = \frac{t}{t ^{2} + 16}

= \frac{1 \cdot t ^{2} + 16 - \frac{t}{t ^{2} + 16} t}{( t ^{2} + 16 ) ^{2}}

簡素化

\frac{1 \cdot t ^{2} + 16 - \frac{t}{t ^{2} + 16} t}{( t ^{2} + 16 ) ^{2}} : \frac{16}{( t ^{2} + 16 ) t ^{2} + 16}

= \frac{16}{( t ^{2} + 16 ) t ^{2} + 16}