derivative (1-2x^2)/(x^4-2x^2+6)

解答

导数

\frac{1 - 2 x ^{2}}{x ^{4} - 2 x ^{2} + 6}

\frac{4 x ^{5} - 4 x ^{3} - 20 x}{( x ^{4} - 2 x ^{2} + 6 ) ^{2}}

求解步骤

\frac{d}{d x} (\frac{1 - 2 x ^{2}}{x ^{4} - 2 x ^{2} + 6})

使用微分除法定则:

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

= \frac{\frac{d}{d x} ( 1 - 2 x ^{2} ) ( x ^{4} - 2 x ^{2} + 6 ) - \frac{d}{d x} ( x ^{4} - 2 x ^{2} + 6 ) ( 1 - 2 x ^{2} )}{( x ^{4} - 2 x ^{2} + 6 ) ^{2}}

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

\frac{d}{d x} (x^{4} - 2 x^{2} + 6) = 4 x^{3} - 4 x

= \frac{( - 4 x ) ( x ^{4} - 2 x ^{2} + 6 ) - ( 4 x ^{3} - 4 x ) ( 1 - 2 x ^{2} )}{( x ^{4} - 2 x ^{2} + 6 ) ^{2}}

化简

\frac{( - 4 x ) ( x ^{4} - 2 x ^{2} + 6 ) - ( 4 x ^{3} - 4 x ) ( 1 - 2 x ^{2} )}{( x ^{4} - 2 x ^{2} + 6 ) ^{2}} : \frac{4 x ^{5} - 4 x ^{3} - 20 x}{( x ^{4} - 2 x ^{2} + 6 ) ^{2}}

= \frac{4 x ^{5} - 4 x ^{3} - 20 x}{( x ^{4} - 2 x ^{2} + 6 ) ^{2}}

h \to 0 lim (\frac{( f ( h ) ( 0 + h ) - f ( 0 ) )}{h})

\int x f (x) d x

f (x) = 17 e^{x}

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

\frac{x d y}{d x} + 5 y = x^{3} - x