derivative F(x)=5x^5(x^3-9x)

解答

导数

F (x) = 5 x^{5} (x^{3} - 9 x)

5 (8 x^{7} - 54 x^{5})

求解步骤

\frac{d}{d x} (5 x^{5} (x^{3} - 9 x))

将常数提出:

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

= 5 \frac{d}{d x} (x^{5} (x^{3} - 9 x))

使用微分乘法定则:

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

f = x^{5}, g = x^{3} - 9 x

= 5 (\frac{d}{d x} (x^{5}) (x^{3} - 9 x) + \frac{d}{d x} (x^{3} - 9 x) x^{5})

\frac{d}{d x} (x^{5}) = 5 x^{4}

\frac{d}{d x} (x^{3} - 9 x) = 3 x^{2} - 9

= 5 (5 x^{4} (x^{3} - 9 x) + (3 x^{2} - 9) x^{5})

乘开

5 x^{4} (x^{3} - 9 x) + (3 x^{2} - 9) x^{5} : 8 x^{7} - 54 x^{5}

= 5 (8 x^{7} - 54 x^{5})

x \to \infty lim (\frac{2 x + 1}{2})

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

\int 2 x sin (4 x) d x

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\frac{\partial}{\partial x} (\frac{a}{x})