derivative of (cos(x)^x)

解

\frac{d}{d x} ((cos (x))^{x})

cos^{x} (x) (ln (cos (x)) - x tan (x))

解答ステップ

\frac{d}{d x} ((cos (x))^{x})

指数の規則を適用する:

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

(cos (x))^{x} = e^{x l n (c o s (x))}

= \frac{d}{d x} (e^{x l n (c o s (x))})

連鎖法則を適用:

e^{x l n (c o s (x))} \frac{d}{d x} (x ln (cos (x)))

= e^{x l n (c o s (x))} \frac{d}{d x} (x ln (cos (x)))

\frac{d}{d x} (x ln (cos (x))) = ln (cos (x)) - x tan (x)

= e^{x l n (c o s (x))} (ln (cos (x)) - x tan (x))

e^{x l n (c o s (x))} = cos^{x} (x)

= cos^{x} (x) (ln (cos (x)) - x tan (x))