derivative f(x)=(x^2-6x+1)/(x^2+4)

解

導関数

f (x) = \frac{x ^{2} - 6 x + 1}{x ^{2} + 4}

\frac{6 x ^{2} + 6 x - 24}{( x ^{2} + 4 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x ^{2} - 6 x + 1}{x ^{2} + 4})

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 ^{2} - 6 x + 1 ) ( x ^{2} + 4 ) - \frac{d}{d x} ( x ^{2} + 4 ) ( x ^{2} - 6 x + 1 )}{( x ^{2} + 4 ) ^{2}}

\frac{d}{d x} (x^{2} - 6 x + 1) = 2 x - 6

\frac{d}{d x} (x^{2} + 4) = 2 x

= \frac{( 2 x - 6 ) ( x ^{2} + 4 ) - 2 x ( x ^{2} - 6 x + 1 )}{( x ^{2} + 4 ) ^{2}}

拡張

(2 x - 6) (x^{2} + 4) - 2 x (x^{2} - 6 x + 1) : 6 x^{2} + 6 x - 24

= \frac{6 x ^{2} + 6 x - 24}{( x ^{2} + 4 ) ^{2}}