derivative f(x)=(x^2)/(x^2-16)

解

導関数

f (x) = \frac{x ^{2}}{x ^{2} - 16}

- \frac{32 x}{( x ^{2} - 16 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x ^{2}}{x ^{2} - 16})

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} ) ( x ^{2} - 16 ) - \frac{d}{d x} ( x ^{2} - 16 ) x ^{2}}{( x ^{2} - 16 ) ^{2}}

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

\frac{d}{d x} (x^{2} - 16) = 2 x

= \frac{2 x ( x ^{2} - 16 ) - 2 x x ^{2}}{( x ^{2} - 16 ) ^{2}}

簡素化

\frac{2 x ( x ^{2} - 16 ) - 2 x x ^{2}}{( x ^{2} - 16 ) ^{2}} : - \frac{32 x}{( x ^{2} - 16 ) ^{2}}

= - \frac{32 x}{( x ^{2} - 16 ) ^{2}}