d/(da)((pi(9a^2-1))/(54a^2))

解答

\frac{d}{d a} (\frac{π ( 9 a ^{2} - 1 )}{54 a ^{2}})

\frac{π}{27 a ^{3}}

求解步骤

\frac{d}{d a} (\frac{π ( 9 a ^{2} - 1 )}{54 a ^{2}})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= \frac{π}{54} \frac{d}{d a} (\frac{9 a ^{2} - 1}{a ^{2}})

使用微分除法定则:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{π}{54} \cdot \frac{\frac{d}{d a} ( 9 a ^{2} - 1 ) a ^{2} - \frac{d}{d a} ( a ^{2} ) ( 9 a ^{2} - 1 )}{( a ^{2} ) ^{2}}

\frac{d}{d a} (9 a^{2} - 1) = 18 a

\frac{d}{d a} (a^{2}) = 2 a

= \frac{π}{54} \cdot \frac{18 a a ^{2} - 2 a ( 9 a ^{2} - 1 )}{( a ^{2} ) ^{2}}

化简

\frac{π}{54} \cdot \frac{18 a a ^{2} - 2 a ( 9 a ^{2} - 1 )}{( a ^{2} ) ^{2}} : \frac{π}{27 a ^{3}}

= \frac{π}{27 a ^{3}}

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