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

解

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

sin (2 x) csc (x^{2}) - 2 x sin^{2} (x) cot (x^{2}) csc (x^{2})

解答ステップ

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

簡素化

\frac{sin ^{2} ( x )}{sin ( x ^{2} )} : sin^{2} (x) csc (x^{2})

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

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = sin^{2} (x), g = csc (x^{2})

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

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

\frac{d}{d x} (csc (x^{2})) = - cot (x^{2}) csc (x^{2}) \cdot 2 x

= sin (2 x) csc (x^{2}) + (- cot (x^{2}) csc (x^{2}) \cdot 2 x) sin^{2} (x)

簡素化

= sin (2 x) csc (x^{2}) - 2 x sin^{2} (x) cot (x^{2}) csc (x^{2})