derivative (sin(x))/((sin(x)+x^2))

解

導関数

\frac{sin ( x )}{( sin ( x ) + x ^{2} )}

\frac{x ^{2} cos ( x ) - 2 x sin ( x )}{( sin ( x ) + x ^{2} ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{sin ( x )}{sin ( x ) + x ^{2}})

商の法則を適用:

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

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

\frac{d}{d x} (sin (x)) = cos (x)

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

= \frac{cos ( x ) ( sin ( x ) + x ^{2} ) - ( cos ( x ) + 2 x ) sin ( x )}{( sin ( x ) + x ^{2} ) ^{2}}

拡張

cos (x) (sin (x) + x^{2}) - (cos (x) + 2 x) sin (x) : x^{2} cos (x) - 2 x sin (x)

= \frac{x ^{2} cos ( x ) - 2 x sin ( x )}{( sin ( x ) + x ^{2} ) ^{2}}