derivative (x^2)/(6+sqrt(x))

解答

导数

\frac{x ^{2}}{6 + x}

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

求解步骤

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

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

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

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

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

化简

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

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

x \to - 1 lim (- x^{2} + 4 x - 5)

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

t an g e n t f (x) = 4 x^{2} - 2 x + 4, at x = 0

t an g e n t f (x) = - 2 x^{4} + 5 x^{2} - 3, (1, 0)

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