(\partial)/(\partial x)((x^2-x)/((L+2x)^3))

解答

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

\frac{- 2 x ^{2} + 2 Lx + 4 x - L}{( L + 2 x ) ^{4}}

求解步骤

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

使用微分除法定则:

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

= \frac{\frac{\partial}{\partial x} ( x ^{2} - x ) ( L + 2 x ) ^{3} - \frac{\partial}{\partial x} ( ( L + 2 x ) ^{3} ) ( x ^{2} - x )}{( ( L + 2 x ) ^{3} ) ^{2}}

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

\frac{\partial}{\partial x} ((L + 2 x)^{3}) = 6 (L + 2 x)^{2}

= \frac{( 2 x - 1 ) ( L + 2 x ) ^{3} - 6 ( L + 2 x ) ^{2} ( x ^{2} - x )}{( ( L + 2 x ) ^{3} ) ^{2}}

化简

\frac{( 2 x - 1 ) ( L + 2 x ) ^{3} - 6 ( L + 2 x ) ^{2} ( x ^{2} - x )}{( ( L + 2 x ) ^{3} ) ^{2}} : \frac{- 2 x ^{2} + 2 Lx + 4 x - L}{( L + 2 x ) ^{4}}

= \frac{- 2 x ^{2} + 2 Lx + 4 x - L}{( L + 2 x ) ^{4}}

x \to + 2 - lim (x + c)

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