derivative of (11-3x^{2/3}(x-2)^4)

解答

\frac{d}{d x} ((11 - 3 x)^{\frac{2}{3}} (x - 2)^{4})

\frac{- 2 ( x - 2 ) ^{4} + 4 ( x - 2 ) ^{3} ( 11 - 3 x )}{( 11 - 3 x ) ^{\frac{1}{3}}}

求解步骤

\frac{d}{d x} ((11 - 3 x)^{\frac{2}{3}} (x - 2)^{4})

使用微分乘法定则:

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

f = (11 - 3 x)^{\frac{2}{3}}, g = (x - 2)^{4}

= \frac{d}{d x} ((11 - 3 x)^{\frac{2}{3}}) (x - 2)^{4} + \frac{d}{d x} ((x - 2)^{4}) (11 - 3 x)^{\frac{2}{3}}

\frac{d}{d x} ((11 - 3 x)^{\frac{2}{3}}) = - \frac{2}{( 11 - 3 x ) ^{\frac{1}{3}}}

\frac{d}{d x} ((x - 2)^{4}) = 4 (x - 2)^{3}

= (- \frac{2}{( 11 - 3 x ) ^{\frac{1}{3}}}) (x - 2)^{4} + 4 (x - 2)^{3} (11 - 3 x)^{\frac{2}{3}}

化简

(- \frac{2}{( 11 - 3 x ) ^{\frac{1}{3}}}) (x - 2)^{4} + 4 (x - 2)^{3} (11 - 3 x)^{\frac{2}{3}} : \frac{- 2 ( x - 2 ) ^{4} + 4 ( x - 2 ) ^{3} ( 11 - 3 x )}{( 11 - 3 x ) ^{\frac{1}{3}}}

= \frac{- 2 ( x - 2 ) ^{4} + 4 ( x - 2 ) ^{3} ( 11 - 3 x )}{( 11 - 3 x ) ^{\frac{1}{3}}}

\int \frac{x ^{2} + 2 x + 3}{( x - 1 ) ( x + 1 ) ^{2}} d x

x \to 0 lim ((sin (x))^{2})

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