derivative y=(1+ln(x))^{sin(x)}

解

導関数

y = (1 + ln (x))^{s i n (x)}

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

解答ステップ

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

指数の規則を適用する:

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

(1 + ln (x))^{s i n (x)} = e^{s i n (x) l n (1 + l n (x))}

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

連鎖法則を適用:

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

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

\frac{d}{d x} (sin (x) ln (1 + ln (x))) = cos (x) ln (1 + ln (x)) + \frac{sin ( x )}{x ( 1 + ln ( x ) )}

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

e^{s i n (x) l n (1 + l n (x))} = (1 + ln (x))^{s i n (x)}

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