derivative f(x)=(x^{1/3}-4)/(x^{7/2)}

解答

导数

f (x) = \frac{x ^{\frac{1}{3}} - 4}{x ^{\frac{7}{2}}}

\frac{- 19 x ^{\frac{17}{6}} + 84 x ^{\frac{5}{2}}}{6 x ^{7}}

求解步骤

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

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

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

\frac{d}{d x} (x^{\frac{7}{2}}) = \frac{7 x ^{\frac{5}{2}}}{2}

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

化简

\frac{\frac{1}{3 x ^{\frac{2}{3}}} x ^{\frac{7}{2}} - \frac{7 x ^{\frac{5}{2}}}{2} ( x ^{\frac{1}{3}} - 4 )}{( x ^{\frac{7}{2}} ) ^{2}} : \frac{- 19 x ^{\frac{17}{6}} + 84 x ^{\frac{5}{2}}}{6 x ^{7}}

= \frac{- 19 x ^{\frac{17}{6}} + 84 x ^{\frac{5}{2}}}{6 x ^{7}}

d er i v a t i v e \frac{3}{4} s^{2}

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

z \to 1 + i lim (z^{2} + i)

\int (3 x - 5)^{4} d x

d er i v a t i v e f (x) = 5 x^{2} + 3 x + 8