derivative f(x)=sqrt(x/(x+5))

解

導関数

f (x) = \frac{x}{x + 5}

\frac{5}{2 x ( x + 5 ) ^{\frac{3}{2}}}

解答ステップ

\frac{d}{d x} (\frac{x}{x + 5})

連鎖法則を適用:

\frac{1}{2 \frac{x}{x + 5}} \frac{d}{d x} (\frac{x}{x + 5})

= \frac{1}{2 \frac{x}{x + 5}} \frac{d}{d x} (\frac{x}{x + 5})

\frac{d}{d x} (\frac{x}{x + 5}) = \frac{5}{( x + 5 ) ^{2}}

= \frac{1}{2 \frac{x}{x + 5}} \cdot \frac{5}{( x + 5 ) ^{2}}

簡素化

\frac{1}{2 \frac{x}{x + 5}} \cdot \frac{5}{( x + 5 ) ^{2}} : \frac{5}{2 x ( x + 5 ) ^{\frac{3}{2}}}

= \frac{5}{2 x ( x + 5 ) ^{\frac{3}{2}}}