derivative of ln(e^{sin(x}*x^{cos(x)}))

解

\frac{d}{d x} (ln (e^{s i n (x)} \cdot x^{c o s (x)}))

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

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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