derivative of 7sqrt(x)(x^3-3sqrt(x)+2)

解答

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

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

求解步骤

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

将常数提出:

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

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

使用微分乘法定则:

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

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

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

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

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

= 7 (\frac{1}{2 x} (x^{3} - 3 x + 2) + (3 x^{2} - \frac{3}{2 x}) x)

化简

7 (\frac{1}{2 x} (x^{3} - 3 x + 2) + (3 x^{2} - \frac{3}{2 x}) x) : 7 (- \frac{3}{2} + \frac{7 x ^{3} - 3 x + 2}{2 x})

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