derivative of x^3(ln(x)^{cos(x)})

解

\frac{d}{d x} (x^{3} (ln (x))^{c o s (x)})

3 x^{2} ln^{c o s (x)} (x) + x^{3} ln^{c o s (x)} (x) (- sin (x) ln (ln (x)) + \frac{cos ( x )}{x ln ( x )})

解答ステップ

\frac{d}{d x} (x^{3} (ln (x))^{c o s (x)})

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{3}, g = (ln (x))^{c o s (x)}

= \frac{d}{d x} (x^{3}) (ln (x))^{c o s (x)} + \frac{d}{d x} ((ln (x))^{c o s (x)}) x^{3}

\frac{d}{d x} (x^{3}) = 3 x^{2}

\frac{d}{d x} ((ln (x))^{c o s (x)}) = ln^{c o s (x)} (x) (- sin (x) ln (ln (x)) + \frac{cos ( x )}{x ln ( x )})

= 3 x^{2} (ln (x))^{c o s (x)} + ln^{c o s (x)} (x) (- sin (x) ln (ln (x)) + \frac{cos ( x )}{x ln ( x )}) x^{3}

簡素化

= 3 x^{2} ln^{c o s (x)} (x) + x^{3} ln^{c o s (x)} (x) (- sin (x) ln (ln (x)) + \frac{cos ( x )}{x ln ( x )})