derivative (2x+1)/((x^2+1)^{1.5)}

解

導関数

\frac{2 x + 1}{( x ^{2} + 1 ) ^{1.5}}

\frac{2 ( x ^{2} + 1 ) ^{1.5} - 3 x ( 2 x + 1 ) x ^{2} + 1}{( x ^{2} + 1 ) ^{3}}

解答ステップ

\frac{d}{d x} (\frac{2 x + 1}{( x ^{2} + 1 ) ^{1.5}})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( 2 x + 1 ) ( x ^{2} + 1 ) ^{1.5} - \frac{d}{d x} ( ( x ^{2} + 1 ) ^{1.5} ) ( 2 x + 1 )}{( ( x ^{2} + 1 ) ^{1.5} ) ^{2}}

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

\frac{d}{d x} ((x^{2} + 1)^{1.5}) = 3 x x^{2} + 1

= \frac{2 ( x ^{2} + 1 ) ^{1.5} - 3 x x ^{2} + 1 ( 2 x + 1 )}{( ( x ^{2} + 1 ) ^{1.5} ) ^{2}}

((x^{2} + 1)^{1.5})^{2} = (x^{2} + 1)^{3}

= \frac{2 ( x ^{2} + 1 ) ^{1.5} - 3 x ( 2 x + 1 ) x ^{2} + 1}{( x ^{2} + 1 ) ^{3}}