derivative y=(t-sqrt(t))/(t^{1/3)}

解答

导数

y = \frac{t - t}{t ^{\frac{1}{3}}}

\frac{t - 1}{6 t ^{\frac{5}{6}}} + \frac{1}{2 t ^{\frac{1}{3}}}

求解步骤

\frac{d}{d t} (\frac{t - t}{t ^{\frac{1}{3}}})

化简

\frac{t - t}{t ^{\frac{1}{3}}} : t^{\frac{1}{6}} (t - 1)

= \frac{d}{d t} (t^{\frac{1}{6}} (t - 1))

使用微分乘法定则:

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

f = t^{\frac{1}{6}}, g = t - 1

= \frac{d}{d t} (t^{\frac{1}{6}}) (t - 1) + \frac{d}{d t} (t - 1) t^{\frac{1}{6}}

\frac{d}{d t} (t^{\frac{1}{6}}) = \frac{1}{6 t ^{\frac{5}{6}}}

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

= \frac{1}{6 t ^{\frac{5}{6}}} (t - 1) + \frac{1}{2 t ^{\frac{1}{2}}} t^{\frac{1}{6}}

化简

\frac{1}{6 t ^{\frac{5}{6}}} (t - 1) + \frac{1}{2 t ^{\frac{1}{2}}} t^{\frac{1}{6}} : \frac{t - 1}{6 t ^{\frac{5}{6}}} + \frac{1}{2 t ^{\frac{1}{3}}}

= \frac{t - 1}{6 t ^{\frac{5}{6}}} + \frac{1}{2 t ^{\frac{1}{3}}}

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