derivative of (5x^4+7x^2-3/(2x^2))

解答

\frac{d}{d x} (\frac{5 x ^{4} + 7 x ^{2} - 3}{2 x ^{2}})

\frac{5 x ^{4} + 3}{x ^{3}}

求解步骤

\frac{d}{d x} (\frac{5 x ^{4} + 7 x ^{2} - 3}{2 x ^{2}})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= \frac{1}{2} \frac{d}{d x} (\frac{5 x ^{4} + 7 x ^{2} - 3}{x ^{2}})

使用微分除法定则:

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

= \frac{1}{2} \cdot \frac{\frac{d}{d x} ( 5 x ^{4} + 7 x ^{2} - 3 ) x ^{2} - \frac{d}{d x} ( x ^{2} ) ( 5 x ^{4} + 7 x ^{2} - 3 )}{( x ^{2} ) ^{2}}

\frac{d}{d x} (5 x^{4} + 7 x^{2} - 3) = 20 x^{3} + 14 x

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

= \frac{1}{2} \cdot \frac{( 20 x ^{3} + 14 x ) x ^{2} - 2 x ( 5 x ^{4} + 7 x ^{2} - 3 )}{( x ^{2} ) ^{2}}

化简

\frac{1}{2} \cdot \frac{( 20 x ^{3} + 14 x ) x ^{2} - 2 x ( 5 x ^{4} + 7 x ^{2} - 3 )}{( x ^{2} ) ^{2}} : \frac{5 x ^{4} + 3}{x ^{3}}

= \frac{5 x ^{4} + 3}{x ^{3}}

\int 4 x e^{x^{2}} d x

\int 3 y e^{3 z} d z

\int (x + \frac{1}{8 x}) d x

\int \frac{12}{x} d x

d er i v a t i v e ln (x^{4} + x)