derivative of (cos(4x)^x)

解

\frac{d}{d x} ((cos (4 x))^{x})

cos^{x} (4 x) (ln (cos (4 x)) - 4 x tan (4 x))

解答ステップ

\frac{d}{d x} ((cos (4 x))^{x})

指数の規則を適用する:

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

(cos (4 x))^{x} = e^{x l n (c o s (4 x))}

= \frac{d}{d x} (e^{x l n (c o s (4 x))})

連鎖法則を適用:

e^{x l n (c o s (4 x))} \frac{d}{d x} (x ln (cos (4 x)))

= e^{x l n (c o s (4 x))} \frac{d}{d x} (x ln (cos (4 x)))

\frac{d}{d x} (x ln (cos (4 x))) = ln (cos (4 x)) - 4 x tan (4 x)

= e^{x l n (c o s (4 x))} (ln (cos (4 x)) - 4 x tan (4 x))

e^{x l n (c o s (4 x))} = cos^{x} (4 x)

= cos^{x} (4 x) (ln (cos (4 x)) - 4 x tan (4 x))