derivative of ln((sin(x)/x))

解

\frac{d}{d x} (ln (\frac{sin ( x )}{x}))

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

解答ステップ

\frac{d}{d x} (ln (\frac{sin ( x )}{x}))

連鎖法則を適用:

\frac{1}{\frac{s i n ( x )}{x}} \frac{d}{d x} (\frac{sin ( x )}{x})

= \frac{1}{\frac{s i n ( x )}{x}} \frac{d}{d x} (\frac{sin ( x )}{x})

\frac{d}{d x} (\frac{sin ( x )}{x}) = \frac{x cos ( x ) - sin ( x )}{x ^{2}}

= \frac{1}{\frac{s i n ( x )}{x}} \cdot \frac{x cos ( x ) - sin ( x )}{x ^{2}}

簡素化

\frac{1}{\frac{s i n ( x )}{x}} \cdot \frac{x cos ( x ) - sin ( x )}{x ^{2}} : \frac{x cos ( x ) - sin ( x )}{x sin ( x )}

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