derivative f(x)=7x^4arctan(4x^4)

解答

导数

f (x) = 7 x^{4} arctan (4 x^{4})

7 (4 x^{3} arctan (4 x^{4}) + \frac{16 x ^{7}}{16 x ^{8} + 1})

求解步骤

\frac{d}{d x} (7 x^{4} arctan (4 x^{4}))

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 7 \frac{d}{d x} (x^{4} arctan (4 x^{4}))

使用微分乘法定则:

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

f = x^{4}, g = arctan (4 x^{4})

= 7 (\frac{d}{d x} (x^{4}) arctan (4 x^{4}) + \frac{d}{d x} (arctan (4 x^{4})) x^{4})

\frac{d}{d x} (x^{4}) = 4 x^{3}

\frac{d}{d x} (arctan (4 x^{4})) = \frac{16 x ^{3}}{16 x ^{8} + 1}

= 7 (4 x^{3} arctan (4 x^{4}) + \frac{16 x ^{3}}{16 x ^{8} + 1} x^{4})

\frac{16 x ^{3}}{16 x ^{8} + 1} x^{4} = \frac{16 x ^{7}}{16 x ^{8} + 1}

= 7 (4 x^{3} arctan (4 x^{4}) + \frac{16 x ^{7}}{16 x ^{8} + 1})

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