derivative x(x^2+7)^4

解答

导数

x (x^{2} + 7)^{4}

(x^{2} + 7)^{4} + 8 x^{2} (x^{2} + 7)^{3}

求解步骤

\frac{d}{d x} (x (x^{2} + 7)^{4})

使用微分乘法定则:

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

f = x, g = (x^{2} + 7)^{4}

= \frac{d x}{d x} (x^{2} + 7)^{4} + \frac{d}{d x} ((x^{2} + 7)^{4}) x

\frac{d x}{d x} = 1

\frac{d}{d x} ((x^{2} + 7)^{4}) = 8 x (x^{2} + 7)^{3}

= 1 \cdot (x^{2} + 7)^{4} + 8 x (x^{2} + 7)^{3} x

化简

1 \cdot (x^{2} + 7)^{4} + 8 x (x^{2} + 7)^{3} x : (x^{2} + 7)^{4} + 8 x^{2} (x^{2} + 7)^{3}

= (x^{2} + 7)^{4} + 8 x^{2} (x^{2} + 7)^{3}

t an g e n t 2 x^{3} - 2 x^{2} + 1

x \to 7 + lim (\frac{3}{x - 7})

\int_{5}^{\infty} \frac{1}{x ^{2} - 4} d x

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

x \to 2 lim (2 + 3)