derivative of [arctan(x](1+x^2))

解答

\frac{d}{d x} ([arctan (x)] (1 + x^{2}))

1 + 2 x arctan (x)

求解步骤

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

使用微分乘法定则:

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

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

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

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

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

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

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

= 1 + 2 x arctan (x)

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