derivative 2/3 (4y^2+1/2)^{3/2}

解

導関数

\frac{2}{3} (4 y^{2} + \frac{1}{2})^{\frac{3}{2}}

4 \cdot 2^{\frac{1}{2}} y (8 y^{2} + 1)^{\frac{1}{2}}

解答ステップ

\frac{d}{d y} (\frac{2}{3} (4 y^{2} + \frac{1}{2})^{\frac{3}{2}})

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= \frac{2}{3} \frac{d}{d y} ((4 y^{2} + \frac{1}{2})^{\frac{3}{2}})

連鎖法則を適用:

\frac{3 ( 4 y ^{2} + \frac{1}{2} ) ^{\frac{1}{2}}}{2} \frac{d}{d y} (4 y^{2} + \frac{1}{2})

= \frac{3 ( 4 y ^{2} + \frac{1}{2} ) ^{\frac{1}{2}}}{2} \frac{d}{d y} (4 y^{2} + \frac{1}{2})

\frac{d}{d y} (4 y^{2} + \frac{1}{2}) = 8 y

= \frac{2}{3} \cdot \frac{3 ( 4 y ^{2} + \frac{1}{2} ) ^{\frac{1}{2}}}{2} \cdot 8 y

簡素化

\frac{2}{3} \cdot \frac{3 ( 4 y ^{2} + \frac{1}{2} ) ^{\frac{1}{2}}}{2} \cdot 8 y : 4 \cdot 2^{\frac{1}{2}} y (8 y^{2} + 1)^{\frac{1}{2}}

= 4 \cdot 2^{\frac{1}{2}} y (8 y^{2} + 1)^{\frac{1}{2}}