derivative x^2arctan(2x)

解答

导数

x^{2} arctan (2 x)

2 x arctan (2 x) + \frac{2 x ^{2}}{4 x ^{2} + 1}

求解步骤

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

使用微分乘法定则:

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

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

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

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

\frac{d}{d x} (arctan (2 x)) = \frac{2}{4 x ^{2} + 1}

= 2 x arctan (2 x) + \frac{2}{4 x ^{2} + 1} x^{2}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= 2 x arctan (2 x) + \frac{2 x ^{2}}{4 x ^{2} + 1}

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