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

解

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

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

解答ステップ

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

簡素化

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

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

積の法則を適用:

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

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

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

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

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

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

簡素化

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