derivative f(x)=(x^3-1x^2+2)/(x^2)

解

導関数

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

\frac{x ^{3} - 4}{x ^{3}}

解答ステップ

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

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

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

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

簡素化

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

= \frac{x ^{3} - 4}{x ^{3}}