derivative of x^2arctan(ln(x))

解

\frac{d}{d x} (x^{2} arctan (ln (x)))

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

解答ステップ

\frac{d}{d x} (x^{2} arctan (ln (x)))

積の法則を適用:

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

f = x^{2}, g = arctan (ln (x))

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

\frac{d}{d x} (x^{2}) = 2 x

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

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

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

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