derivative of cos(xln(x^2+1))

解

\frac{d}{d x} (cos (x) ln (x^{2} + 1))

- sin (x) ln (x^{2} + 1) + \frac{2 x cos ( x )}{x ^{2} + 1}

解答ステップ

\frac{d}{d x} (cos (x) ln (x^{2} + 1))

積の法則を適用:

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

f = cos (x), g = ln (x^{2} + 1)

= \frac{d}{d x} (cos (x)) ln (x^{2} + 1) + \frac{d}{d x} (ln (x^{2} + 1)) cos (x)

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

\frac{d}{d x} (ln (x^{2} + 1)) = \frac{2 x}{x ^{2} + 1}

= (- sin (x)) ln (x^{2} + 1) + \frac{2 x}{x ^{2} + 1} cos (x)

簡素化

(- sin (x)) ln (x^{2} + 1) + \frac{2 x}{x ^{2} + 1} cos (x) : - sin (x) ln (x^{2} + 1) + \frac{2 x cos ( x )}{x ^{2} + 1}

= - sin (x) ln (x^{2} + 1) + \frac{2 x cos ( x )}{x ^{2} + 1}