derivative arccos(x^{1/2})

解

導関数

arccos (x^{\frac{1}{2}})

- \frac{1}{2 x ^{\frac{1}{2}} 1 - x}

解答ステップ

\frac{d}{d x} (arccos (x^{\frac{1}{2}}))

連鎖法則を適用:

- \frac{1}{1 - ( x ^{\frac{1}{2}} ) ^{2}} \frac{d}{d x} (x^{\frac{1}{2}})

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

\frac{d}{d x} (x^{\frac{1}{2}}) = \frac{1}{2 x ^{\frac{1}{2}}}

= - \frac{1}{1 - ( x ^{\frac{1}{2}} ) ^{2}} \cdot \frac{1}{2 x ^{\frac{1}{2}}}

簡素化

- \frac{1}{1 - ( x ^{\frac{1}{2}} ) ^{2}} \cdot \frac{1}{2 x ^{\frac{1}{2}}} : - \frac{1}{2 x ^{\frac{1}{2}} 1 - x}

= - \frac{1}{2 x ^{\frac{1}{2}} 1 - x}