derivative f(x)=x^9arccos(x)

解答

导数

f (x) = x^{9} arccos (x)

9 x^{8} arccos (x) - \frac{x ^{9}}{1 - x ^{2}}

求解步骤

\frac{d}{d x} (x^{9} arccos (x))

使用微分乘法定则:

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

f = x^{9}, g = arccos (x)

= \frac{d}{d x} (x^{9}) arccos (x) + \frac{d}{d x} (arccos (x)) x^{9}

\frac{d}{d x} (x^{9}) = 9 x^{8}

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

= 9 x^{8} arccos (x) + (- \frac{1}{1 - x ^{2}}) x^{9}

化简

9 x^{8} arccos (x) + (- \frac{1}{1 - x ^{2}}) x^{9} : 9 x^{8} arccos (x) - \frac{x ^{9}}{1 - x ^{2}}

= 9 x^{8} arccos (x) - \frac{x ^{9}}{1 - x ^{2}}

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