derivative of (sin(x)/((x+1)^2))

解

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

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

解答ステップ

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

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

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

= \frac{cos ( x ) ( x + 1 ) ^{2} - 2 ( x + 1 ) sin ( x )}{( ( x + 1 ) ^{2} ) ^{2}}

簡素化

\frac{cos ( x ) ( x + 1 ) ^{2} - 2 ( x + 1 ) sin ( x )}{( ( x + 1 ) ^{2} ) ^{2}} : \frac{cos ( x ) ( x + 1 ) - 2 sin ( x )}{( x + 1 ) ^{3}}

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