derivative f(x)=sqrt(x/(x+9))

解

導関数

f (x) = \frac{x}{x + 9}

\frac{9}{2 x ( x + 9 ) ^{\frac{3}{2}}}

解答ステップ

\frac{d}{d x} (\frac{x}{x + 9})

連鎖法則を適用:

\frac{1}{2 \frac{x}{x + 9}} \frac{d}{d x} (\frac{x}{x + 9})

= \frac{1}{2 \frac{x}{x + 9}} \frac{d}{d x} (\frac{x}{x + 9})

\frac{d}{d x} (\frac{x}{x + 9}) = \frac{9}{( x + 9 ) ^{2}}

= \frac{1}{2 \frac{x}{x + 9}} \cdot \frac{9}{( x + 9 ) ^{2}}

簡素化

\frac{1}{2 \frac{x}{x + 9}} \cdot \frac{9}{( x + 9 ) ^{2}} : \frac{9}{2 x ( x + 9 ) ^{\frac{3}{2}}}

= \frac{9}{2 x ( x + 9 ) ^{\frac{3}{2}}}