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

解

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

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

解答ステップ

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

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

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

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

簡素化

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

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