derivative (cos(9x))^x

解

導関数

(cos (9 x))^{x}

cos^{x} (9 x) (ln (cos (9 x)) - 9 x tan (9 x))

解答ステップ

\frac{d}{d x} ((cos (9 x))^{x})

指数の規則を適用する:

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

(cos (9 x))^{x} = e^{x l n (c o s (9 x))}

= \frac{d}{d x} (e^{x l n (c o s (9 x))})

連鎖法則を適用:

e^{x l n (c o s (9 x))} \frac{d}{d x} (x ln (cos (9 x)))

= e^{x l n (c o s (9 x))} \frac{d}{d x} (x ln (cos (9 x)))

\frac{d}{d x} (x ln (cos (9 x))) = ln (cos (9 x)) - 9 x tan (9 x)

= e^{x l n (c o s (9 x))} (ln (cos (9 x)) - 9 x tan (9 x))

e^{x l n (c o s (9 x))} = cos^{x} (9 x)

= cos^{x} (9 x) (ln (cos (9 x)) - 9 x tan (9 x))