derivative f(x)=(ln(x))/(38+x)

解

導関数

f (x) = \frac{ln ( x )}{38 + x}

\frac{38 + x - x ln ( x )}{x ( 38 + x ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{ln ( x )}{38 + x})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( ln ( x ) ) ( 38 + x ) - \frac{d}{d x} ( 38 + x ) ln ( x )}{( 38 + x ) ^{2}}

\frac{d}{d x} (ln (x)) = \frac{1}{x}

\frac{d}{d x} (38 + x) = 1

= \frac{\frac{1}{x} ( 38 + x ) - 1 \cdot ln ( x )}{( 38 + x ) ^{2}}

簡素化

\frac{\frac{1}{x} ( 38 + x ) - 1 \cdot ln ( x )}{( 38 + x ) ^{2}} : \frac{38 + x - x ln ( x )}{x ( 38 + x ) ^{2}}

= \frac{38 + x - x ln ( x )}{x ( 38 + x ) ^{2}}