derivative of (ln(x^2+1)/(3x+1))

解

\frac{d}{d x} (\frac{ln ( x ^{2} + 1 )}{3 x + 1})

\frac{2 x ( 3 x + 1 ) - 3 ln ( x ^{2} + 1 ) ( x ^{2} + 1 )}{( x ^{2} + 1 ) ( 3 x + 1 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{ln ( x ^{2} + 1 )}{3 x + 1})

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 ^{2} + 1 ) ) ( 3 x + 1 ) - \frac{d}{d x} ( 3 x + 1 ) ln ( x ^{2} + 1 )}{( 3 x + 1 ) ^{2}}

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

\frac{d}{d x} (3 x + 1) = 3

= \frac{\frac{2 x}{x ^{2} + 1} ( 3 x + 1 ) - 3 ln ( x ^{2} + 1 )}{( 3 x + 1 ) ^{2}}

簡素化

\frac{\frac{2 x}{x ^{2} + 1} ( 3 x + 1 ) - 3 ln ( x ^{2} + 1 )}{( 3 x + 1 ) ^{2}} : \frac{2 x ( 3 x + 1 ) - 3 ln ( x ^{2} + 1 ) ( x ^{2} + 1 )}{( x ^{2} + 1 ) ( 3 x + 1 ) ^{2}}

= \frac{2 x ( 3 x + 1 ) - 3 ln ( x ^{2} + 1 ) ( x ^{2} + 1 )}{( x ^{2} + 1 ) ( 3 x + 1 ) ^{2}}