derivative of x^{-1}(x^2+1sqrt(x))

解答

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

\frac{3 x ^{2} - 1}{2 x ^{\frac{3}{2}}}

求解步骤

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

使用微分乘法定则:

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

f = x^{- 1}, g = (x^{2} + 1) x

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

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

\frac{d}{d x} ((x^{2} + 1) x) = \frac{5 x ^{2} + 1}{2 x}

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

化简

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

= \frac{3 x ^{2} - 1}{2 x ^{\frac{3}{2}}}

2 y^{^{''}} - 3 y^{^{'}} - 14 y = 0

\frac{d y}{d x} = 0.15 y + 200

(- x)^{^{'}}

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

\int 2 x - 2 x^{2} d x