derivative 1/((t^2+3t+1)^{5/2)}

解答

导数

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

- \frac{5 ( 2 t + 3 )}{2 ( t ^{2} + 3 t + 1 ) ^{\frac{7}{2}}}

求解步骤

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

使用指数法则:

\frac{1}{a} = a^{- 1}

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

使用连锁律:

- \frac{5}{2 ( t ^{2} + 3 t + 1 ) ^{\frac{7}{2}}} \frac{d}{d t} (t^{2} + 3 t + 1)

= - \frac{5}{2 ( t ^{2} + 3 t + 1 ) ^{\frac{7}{2}}} \frac{d}{d t} (t^{2} + 3 t + 1)

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

= - \frac{5}{2 ( t ^{2} + 3 t + 1 ) ^{\frac{7}{2}}} (2 t + 3)

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{5 ( 2 t + 3 )}{2 ( t ^{2} + 3 t + 1 ) ^{\frac{7}{2}}}

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