derivative x/((x^2+sqrt(3x))^5)

解答

导数

\frac{x}{( x ^{2} + 3 x ) ^{5}}

\frac{( x ^{2} + 3 x ) ^{5} - 5 x ( x ^{2} + 3 x ) ^{4} ( 2 x + \frac{3}{2 x} )}{( x ^{2} + 3 x ) ^{10}}

求解步骤

\frac{d}{d x} (\frac{x}{( x ^{2} + 3 x ) ^{5}})

使用微分除法定则:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d x}{d x} ( x ^{2} + 3 x ) ^{5} - \frac{d}{d x} ( ( x ^{2} + 3 x ) ^{5} ) x}{( ( x ^{2} + 3 x ) ^{5} ) ^{2}}

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

\frac{d}{d x} ((x^{2} + 3 x)^{5}) = 5 (x^{2} + 3 x)^{4} (2 x + \frac{3}{2 x})

= \frac{1 \cdot ( x ^{2} + 3 x ) ^{5} - 5 ( x ^{2} + 3 x ) ^{4} ( 2 x + \frac{3}{2 x} ) x}{( ( x ^{2} + 3 x ) ^{5} ) ^{2}}

化简

\frac{1 \cdot ( x ^{2} + 3 x ) ^{5} - 5 ( x ^{2} + 3 x ) ^{4} ( 2 x + \frac{3}{2 x} ) x}{( ( x ^{2} + 3 x ) ^{5} ) ^{2}} : \frac{( x ^{2} + 3 x ) ^{5} - 5 x ( x ^{2} + 3 x ) ^{4} ( 2 x + \frac{3}{2 x} )}{( x ^{2} + 3 x ) ^{10}}

= \frac{( x ^{2} + 3 x ) ^{5} - 5 x ( x ^{2} + 3 x ) ^{4} ( 2 x + \frac{3}{2 x} )}{( x ^{2} + 3 x ) ^{10}}

x \to \infty lim (x n (x))

t an g e n t f (x) = x^{4} - 3 x^{2} + 2, at x = 1

\frac{d}{d t} (\frac{4}{3} π t^{3})

\int (sin^{2} (2 x)) (cos^{4} (2 x)) d x

d er i v a t i v e x^{2} - 8