derivative of (4x^2+9(8x+x^{1/2}))

解答

\frac{d}{d x} ((4 x^{2} + 9) (8 x + x^{\frac{1}{2}}))

\frac{20 x ^{2} + 192 x ^{\frac{5}{2}} + 9}{2 x ^{\frac{1}{2}}} + 72

求解步骤

\frac{d}{d x} ((4 x^{2} + 9) (8 x + x^{\frac{1}{2}}))

使用微分乘法定则:

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

f = 4 x^{2} + 9, g = 8 x + x^{\frac{1}{2}}

= \frac{d}{d x} (4 x^{2} + 9) (8 x + x^{\frac{1}{2}}) + \frac{d}{d x} (8 x + x^{\frac{1}{2}}) (4 x^{2} + 9)

\frac{d}{d x} (4 x^{2} + 9) = 8 x

\frac{d}{d x} (8 x + x^{\frac{1}{2}}) = 8 + \frac{1}{2 x ^{\frac{1}{2}}}

= 8 x (8 x + x^{\frac{1}{2}}) + (8 + \frac{1}{2 x ^{\frac{1}{2}}}) (4 x^{2} + 9)

化简

8 x (8 x + x^{\frac{1}{2}}) + (8 + \frac{1}{2 x ^{\frac{1}{2}}}) (4 x^{2} + 9) : \frac{20 x ^{2} + 192 x ^{\frac{5}{2}} + 9}{2 x ^{\frac{1}{2}}} + 72

= \frac{20 x ^{2} + 192 x ^{\frac{5}{2}} + 9}{2 x ^{\frac{1}{2}}} + 72

d er i v a t i v e y = 7 x^{\frac{1}{2}} + x^{\frac{3}{2}}

\int \frac{5 x}{3 x ^{2} + 4} d x

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\frac{\partial}{\partial y} (\frac{y ^{3}}{x ^{2} + y ^{2}})

n = 0 \sum \infty \frac{3 n ^{9}}{n !}