derivative of (x^2+4x^{1/2}/(x^2))

解

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

- \frac{6}{x ^{\frac{5}{2}}}

解答ステップ

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

商の法則を適用:

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

= \frac{\frac{d}{d x} ( x ^{2} + 4 x ^{\frac{1}{2}} ) x ^{2} - \frac{d}{d x} ( x ^{2} ) ( x ^{2} + 4 x ^{\frac{1}{2}} )}{( x ^{2} ) ^{2}}

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

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

= \frac{\frac{2 + 2 x ^{\frac{3}{2}}}{x ^{\frac{1}{2}}} x ^{2} - 2 x ( x ^{2} + 4 x ^{\frac{1}{2}} )}{( x ^{2} ) ^{2}}

簡素化

\frac{\frac{2 + 2 x ^{\frac{3}{2}}}{x ^{\frac{1}{2}}} x ^{2} - 2 x ( x ^{2} + 4 x ^{\frac{1}{2}} )}{( x ^{2} ) ^{2}} : - \frac{6}{x ^{\frac{5}{2}}}

= - \frac{6}{x ^{\frac{5}{2}}}