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

解

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

- \frac{ln ( 2 ) sin ( 2 x )}{2 ^{s i n (x) + 1} sin ( x )}

解答ステップ

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

指数の規則を適用する:

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

(\frac{1}{2})^{s i n^{2} (x)} = e^{s i n^{2} (x) l n (\frac{1}{2})}

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

連鎖法則を適用:

e^{s i n^{2} (x) l n (\frac{1}{2})} \frac{d}{d x} (sin^{2} (x) ln (\frac{1}{2}))

= e^{s i n^{2} (x) l n (\frac{1}{2})} \frac{d}{d x} (sin^{2} (x) ln (\frac{1}{2}))

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

= e^{s i n^{2} (x) l n (\frac{1}{2})} - \frac{ln ( 2 ) sin ( 2 x )}{2 sin ^{2} ( x )}

簡素化

e^{s i n^{2} (x) l n (\frac{1}{2})} - \frac{ln ( 2 ) sin ( 2 x )}{2 sin ^{2} ( x )} : - \frac{ln ( 2 ) sin ( 2 x )}{2 ^{s i n (x) + 1} sin ( x )}

= - \frac{ln ( 2 ) sin ( 2 x )}{2 ^{s i n (x) + 1} sin ( x )}