derivative of (1-3x^2^{1/(sin^2(x))})

解

\frac{d}{d x} ((1 - 3 x^{2})^{\frac{1}{s i n ^{2} ( x )}})

(- 2 csc^{2} (x) cot (x) ln (1 - 3 x^{2}) - \frac{6 x csc ^{2} ( x )}{1 - 3 x ^{2}}) (1 - 3 x^{2})^{c s c^{2} (x)}

解答ステップ

\frac{d}{d x} ((1 - 3 x^{2})^{\frac{1}{s i n ^{2} ( x )}})

簡素化

(1 - 3 x^{2})^{\frac{1}{s i n ^{2} ( x )}} : (1 - 3 x^{2})^{c s c^{2} (x)}

= \frac{d}{d x} ((1 - 3 x^{2})^{c s c^{2} (x)})

指数の規則を適用する:

a^{b} = e^{b l n (a)}

(1 - 3 x^{2})^{c s c^{2} (x)} = e^{c s c^{2} (x) l n (1 - 3 x^{2})}

= \frac{d}{d x} (e^{c s c^{2} (x) l n (1 - 3 x^{2})})

連鎖法則を適用:

e^{c s c^{2} (x) l n (1 - 3 x^{2})} \frac{d}{d x} (csc^{2} (x) ln (1 - 3 x^{2}))

= e^{c s c^{2} (x) l n (1 - 3 x^{2})} \frac{d}{d x} (csc^{2} (x) ln (1 - 3 x^{2}))

\frac{d}{d x} (csc^{2} (x) ln (1 - 3 x^{2})) = - 2 csc^{2} (x) cot (x) ln (1 - 3 x^{2}) - \frac{6 x csc ^{2} ( x )}{1 - 3 x ^{2}}

= e^{c s c^{2} (x) l n (1 - 3 x^{2})} (- 2 csc^{2} (x) cot (x) ln (1 - 3 x^{2}) - \frac{6 x csc ^{2} ( x )}{1 - 3 x ^{2}})

e^{c s c^{2} (x) l n (1 - 3 x^{2})} = (1 - 3 x^{2})^{c s c^{2} (x)}

= (- 2 csc^{2} (x) cot (x) ln (1 - 3 x^{2}) - \frac{6 x csc ^{2} ( x )}{1 - 3 x ^{2}}) (1 - 3 x^{2})^{c s c^{2} (x)}