derivative f(x)=(x^3+5)/(x^2+1)

解

導関数

f (x) = \frac{x ^{3} + 5}{x ^{2} + 1}

\frac{x ^{4} + 3 x ^{2} - 10 x}{( x ^{2} + 1 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x ^{3} + 5}{x ^{2} + 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} ( x ^{3} + 5 ) ( x ^{2} + 1 ) - \frac{d}{d x} ( x ^{2} + 1 ) ( x ^{3} + 5 )}{( x ^{2} + 1 ) ^{2}}

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

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

= \frac{3 x ^{2} ( x ^{2} + 1 ) - 2 x ( x ^{3} + 5 )}{( x ^{2} + 1 ) ^{2}}

拡張

3 x^{2} (x^{2} + 1) - 2 x (x^{3} + 5) : x^{4} + 3 x^{2} - 10 x

= \frac{x ^{4} + 3 x ^{2} - 10 x}{( x ^{2} + 1 ) ^{2}}