derivative of ln(xcos(x))

解

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

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

解答ステップ

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

積の法則を適用:

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

f = ln (x), g = cos (x)

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

\frac{d}{d x} (ln (x)) = \frac{1}{x}

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

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

簡素化

\frac{1}{x} cos (x) + (- sin (x)) ln (x) : \frac{cos ( x )}{x} - sin (x) ln (x)

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