derivative of (({g}(x)^2)/(h(x)))

解

\frac{d}{d x} (\frac{( g ( x ) ) ^{2}}{h ( x )})

\frac{2 x g ( x ) \frac{d}{d x} ( g ( x ) ) - g ( x ) ^{2}}{x ^{2} h}

解答ステップ

\frac{d}{d x} (\frac{( g ( x ) ) ^{2}}{h ( x )})

h

を定数として扱う

定数を除去:

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

= \frac{1}{h} \frac{d}{d x} (\frac{g ( x ) ^{2}}{x})

商の法則を適用:

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

= \frac{1}{h} \cdot \frac{\frac{d}{d x} ( g ( x ) ^{2} ) x - \frac{d x}{d x} g ( x ) ^{2}}{x ^{2}}

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

\frac{d x}{d x} = 1

= \frac{1}{h} \cdot \frac{2 g ( x ) \frac{d}{d x} ( g ( x ) ) x - 1 \cdot g ( x ) ^{2}}{x ^{2}}

簡素化

\frac{1}{h} \cdot \frac{2 g ( x ) \frac{d}{d x} ( g ( x ) ) x - 1 \cdot g ( x ) ^{2}}{x ^{2}} : \frac{2 x g ( x ) \frac{d}{d x} ( g ( x ) ) - g ( x ) ^{2}}{x ^{2} h}

= \frac{2 x g ( x ) \frac{d}{d x} ( g ( x ) ) - g ( x ) ^{2}}{x ^{2} h}