derivative (cos(x))^{5ln(x)}

解

導関数

(cos (x))^{5 l n (x)}

5 x^{5 l n (c o s (x))} (\frac{ln ( cos ( x ) )}{x} - tan (x) ln (x))

解答ステップ

\frac{d}{d x} ((cos (x))^{5 l n (x)})

指数の規則を適用する:

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

(cos (x))^{5 l n (x)} = e^{5 l n (x) l n (c o s (x))}

= \frac{d}{d x} (e^{5 l n (x) l n (c o s (x))})

連鎖法則を適用:

e^{5 l n (x) l n (c o s (x))} \frac{d}{d x} (5 ln (x) ln (cos (x)))

= e^{5 l n (x) l n (c o s (x))} \frac{d}{d x} (5 ln (x) ln (cos (x)))

\frac{d}{d x} (5 ln (x) ln (cos (x))) = 5 (\frac{ln ( cos ( x ) )}{x} - tan (x) ln (x))

= e^{5 l n (x) l n (c o s (x))} \cdot 5 (\frac{ln ( cos ( x ) )}{x} - tan (x) ln (x))

e^{5 l n (x) l n (c o s (x))} = x^{5 l n (c o s (x))}

= 5 x^{5 l n (c o s (x))} (\frac{ln ( cos ( x ) )}{x} - tan (x) ln (x))