derivative y=ln((5x)/(7+x))

解

導関数

y = ln (\frac{5 x}{7 + x})

\frac{7}{x ( x + 7 )}

解答ステップ

\frac{d}{d x} (ln (\frac{5 x}{7 + x}))

連鎖法則を適用:

\frac{1}{\frac{5 x}{7 + x}} \frac{d}{d x} (\frac{5 x}{7 + x})

= \frac{1}{\frac{5 x}{7 + x}} \frac{d}{d x} (\frac{5 x}{7 + x})

\frac{d}{d x} (\frac{5 x}{7 + x}) = \frac{35}{( 7 + x ) ^{2}}

= \frac{1}{\frac{5 x}{7 + x}} \cdot \frac{35}{( 7 + x ) ^{2}}

簡素化

\frac{1}{\frac{5 x}{7 + x}} \cdot \frac{35}{( 7 + x ) ^{2}} : \frac{7}{x ( x + 7 )}

= \frac{7}{x ( x + 7 )}