derivative of 2pisqrt(x/({g(x))})

解

\frac{d}{d x} (2 π \frac{x}{g ( x )})

\frac{π ( g ( x ) - x \frac{d}{d x} ( g ( x ) ) )}{x g ( x ) ^{\frac{3}{2}}}

解答ステップ

\frac{d}{d x} (2 π \frac{x}{g ( x )})

定数を除去:

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

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

連鎖法則を適用:

\frac{1}{2 \frac{x}{g ( x )}} \frac{d}{d x} (\frac{x}{g ( x )})

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

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

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

簡素化

2 π \frac{1}{2 \frac{x}{g ( x )}} \cdot \frac{g ( x ) - x \frac{d}{d x} ( g ( x ) )}{g ( x ) ^{2}} : \frac{π ( g ( x ) - x \frac{d}{d x} ( g ( x ) ) )}{x g ( x ) ^{\frac{3}{2}}}

= \frac{π ( g ( x ) - x \frac{d}{d x} ( g ( x ) ) )}{x g ( x ) ^{\frac{3}{2}}}