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

解答

\frac{d}{d 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}}

x \to 0 lim (x \cdot co tan (2 π x))

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

y^{^{''}} + 9 y = 0

t an g e n t y = 10 cot (\frac{π x}{20}), at x = 5

\frac{d}{d x} (\frac{3 + x f ( x )}{x})