derivative f(x)=x^{3/2}+1/(x^{5/2)}

解答

导数

f (x) = x^{\frac{3}{2}} + \frac{1}{x ^{\frac{5}{2}}}

\frac{3 x ^{4} - 5}{2 x ^{\frac{7}{2}}}

求解步骤

\frac{d}{d x} (x^{\frac{3}{2}} + \frac{1}{x ^{\frac{5}{2}}})

使用微分加减法定则:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

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

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

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

化简

\frac{3 x ^{\frac{1}{2}}}{2} - \frac{5}{2 x ^{\frac{7}{2}}} : \frac{3 x ^{4} - 5}{2 x ^{\frac{7}{2}}}

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

\int \frac{( e ^{t} + 2 e ^{2 t} )}{e ^{3 t}} d t

\frac{\partial}{\partial x} (\frac{u}{x})

\int_{- 1}^{1} 2 1 - \frac{x ^{2}}{2} d x

t an g e n t \frac{12 x}{x ^{6}}

x \to 1 lim (ln (x^{2}))