derivative f(x)=(sqrt(x)+x)/(x^2)

解

導関数

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

\frac{- 2 x - 3 x}{2 x ^{3}}

解答ステップ

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

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

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

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

簡素化

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

= \frac{- 2 x - 3 x}{2 x ^{3}}