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

解

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

- sin (x) ln (x) + \frac{cos ( x )}{x}

解答ステップ

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

連鎖法則を適用:

\frac{1}{x ^{c o s (x)}} \frac{d}{d x} (x^{c o s (x)})

= \frac{1}{x ^{c o s (x)}} \frac{d}{d x} (x^{c o s (x)})

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

= \frac{1}{x ^{c o s (x)}} x^{c o s (x)} (- sin (x) ln (x) + \frac{cos ( x )}{x})

簡素化

\frac{1}{x ^{c o s (x)}} x^{c o s (x)} (- sin (x) ln (x) + \frac{cos ( x )}{x}) : - sin (x) ln (x) + \frac{cos ( x )}{x}

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