derivative of ((x^2+32x^{1/2})/8)

解

\frac{d}{d x} \frac{( x ^{2} + 32 x ^{\frac{1}{2}} )}{8}

\frac{x ^{\frac{3}{2}} + 8}{4 x ^{\frac{1}{2}}}

解答ステップ

\frac{d}{d x} \frac{( x ^{2} + 32 x ^{\frac{1}{2}} )}{8}

定数を除去:

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

= \frac{1}{8} \frac{d}{d x} (x^{2} + 32 x^{\frac{1}{2}})

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{1}{8} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (32 x^{\frac{1}{2}}))

\frac{d}{d x} (x^{2}) = 2 x

\frac{d}{d x} (32 x^{\frac{1}{2}}) = \frac{16}{x ^{\frac{1}{2}}}

= \frac{1}{8} (2 x + \frac{16}{x ^{\frac{1}{2}}})

簡素化

\frac{1}{8} (2 x + \frac{16}{x ^{\frac{1}{2}}}) : \frac{x ^{\frac{3}{2}} + 8}{4 x ^{\frac{1}{2}}}

= \frac{x ^{\frac{3}{2}} + 8}{4 x ^{\frac{1}{2}}}