derivative f(r)= r/(sqrt(r^2+7))

解

導関数

f (r) = \frac{r}{r ^{2} + 7}

\frac{7}{( r ^{2} + 7 ) r ^{2} + 7}

解答ステップ

\frac{d}{d r} (\frac{r}{r ^{2} + 7})

商の法則を適用:

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

= \frac{\frac{d r}{d r} r ^{2} + 7 - \frac{d}{d r} ( r ^{2} + 7 ) r}{( r ^{2} + 7 ) ^{2}}

\frac{d r}{d r} = 1

\frac{d}{d r} (r^{2} + 7) = \frac{r}{r ^{2} + 7}

= \frac{1 \cdot r ^{2} + 7 - \frac{r}{r ^{2} + 7} r}{( r ^{2} + 7 ) ^{2}}

簡素化

\frac{1 \cdot r ^{2} + 7 - \frac{r}{r ^{2} + 7} r}{( r ^{2} + 7 ) ^{2}} : \frac{7}{( r ^{2} + 7 ) r ^{2} + 7}

= \frac{7}{( r ^{2} + 7 ) r ^{2} + 7}