derivative of ((2x^2+1^3)/(sqrt(x^2-1)))

解答

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

\frac{12 x ( 2 x ^{2} + 1 ) ^{2} ( x ^{2} - 1 ) - x ( 2 x ^{2} + 1 ) ^{3}}{( x ^{2} - 1 ) x ^{2} - 1}

求解步骤

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

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

\frac{d}{d x} ((2 x^{2} + 1)^{3}) = 12 x (2 x^{2} + 1)^{2}

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

= \frac{12 x ( 2 x ^{2} + 1 ) ^{2} x ^{2} - 1 - \frac{x}{x ^{2} - 1} ( 2 x ^{2} + 1 ) ^{3}}{( x ^{2} - 1 ) ^{2}}

化简

\frac{12 x ( 2 x ^{2} + 1 ) ^{2} x ^{2} - 1 - \frac{x}{x ^{2} - 1} ( 2 x ^{2} + 1 ) ^{3}}{( x ^{2} - 1 ) ^{2}} : \frac{12 x ( 2 x ^{2} + 1 ) ^{2} ( x ^{2} - 1 ) - x ( 2 x ^{2} + 1 ) ^{3}}{( x ^{2} - 1 ) x ^{2} - 1}

= \frac{12 x ( 2 x ^{2} + 1 ) ^{2} ( x ^{2} - 1 ) - x ( 2 x ^{2} + 1 ) ^{3}}{( x ^{2} - 1 ) x ^{2} - 1}

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