(\partial)/(\partial x)((x^2)/(sin^2(x)))

解

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

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

解答ステップ

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

簡素化

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

= \frac{\partial}{\partial x} (x^{2} csc^{2} (x))

積の法則を適用:

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

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

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

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

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

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

簡素化

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