derivative f(x)=(x+4)/(\sqrt[4]{x^3)}

解

導関数

f (x) = \frac{x + 4}{4 x ^{3}}

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

解答ステップ

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

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

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

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

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

簡素化

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

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