d/(dt)((t-sqrt(t))/(t^{1/9)})

解答

\frac{d}{d t} (\frac{t - t}{t ^{\frac{1}{9}}})

\frac{7 ( t - 1 )}{18 t ^{\frac{11}{18}}} + \frac{1}{2 t ^{\frac{1}{9}}}

求解步骤

\frac{d}{d t} (\frac{t - t}{t ^{\frac{1}{9}}})

化简

\frac{t - t}{t ^{\frac{1}{9}}} : t^{\frac{7}{18}} (t - 1)

= \frac{d}{d t} (t^{\frac{7}{18}} (t - 1))

使用微分乘法定则:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = t^{\frac{7}{18}}, g = t - 1

= \frac{d}{d t} (t^{\frac{7}{18}}) (t - 1) + \frac{d}{d t} (t - 1) t^{\frac{7}{18}}

\frac{d}{d t} (t^{\frac{7}{18}}) = \frac{7}{18 t ^{\frac{11}{18}}}

\frac{d}{d t} (t - 1) = \frac{1}{2 t ^{\frac{1}{2}}}

= \frac{7}{18 t ^{\frac{11}{18}}} (t - 1) + \frac{1}{2 t ^{\frac{1}{2}}} t^{\frac{7}{18}}

化简

\frac{7}{18 t ^{\frac{11}{18}}} (t - 1) + \frac{1}{2 t ^{\frac{1}{2}}} t^{\frac{7}{18}} : \frac{7 ( t - 1 )}{18 t ^{\frac{11}{18}}} + \frac{1}{2 t ^{\frac{1}{9}}}

= \frac{7 ( t - 1 )}{18 t ^{\frac{11}{18}}} + \frac{1}{2 t ^{\frac{1}{9}}}