derivative of (x^2+7\sqrt[3]{x^2+2})

解答

\frac{d}{d x} ((x^{2} + 7) 3 x^{2} + 2)

2 x 3 x^{2} + 2 + \frac{2 x ( x ^{2} + 7 )}{3 ( x ^{2} + 2 ) ^{\frac{2}{3}}}

求解步骤

\frac{d}{d x} ((x^{2} + 7) 3 x^{2} + 2)

使用微分乘法定则:

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

f = x^{2} + 7, g = 3 x^{2} + 2

= \frac{d}{d x} (x^{2} + 7) 3 x^{2} + 2 + \frac{d}{d x} (3 x^{2} + 2) (x^{2} + 7)

\frac{d}{d x} (x^{2} + 7) = 2 x

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

= 2 x 3 x^{2} + 2 + \frac{2 x}{3 ( x ^{2} + 2 ) ^{\frac{2}{3}}} (x^{2} + 7)

分式相乘:

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

= 2 x 3 x^{2} + 2 + \frac{2 x ( x ^{2} + 7 )}{3 ( x ^{2} + 2 ) ^{\frac{2}{3}}}

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