d/(dM)((2pir^{3/2})/(sqrt(GM)))

解答

\frac{d}{d M} (\frac{2 π r ( M ) ^{\frac{3}{2}}}{G ( M ) M})

\frac{π ( 3 M \frac{d r}{d M} G r ^{\frac{1}{2}} - ( M \frac{d G}{d M} + G ) r ^{\frac{3}{2}} )}{M ^{\frac{3}{2}} G ^{\frac{3}{2}}}

求解步骤

\frac{d}{d M} (\frac{2 π r ^{\frac{3}{2}}}{GM})

将常数提出:

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

= 2 π \frac{d}{d M} (\frac{r ^{\frac{3}{2}}}{GM})

使用微分除法定则:

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

= 2 π \frac{\frac{d}{d M} ( r ^{\frac{3}{2}} ) GM - \frac{d}{d M} ( GM ) r ^{\frac{3}{2}}}{( GM ) ^{2}}

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

\frac{d}{d M} (GM) = \frac{M \frac{d G}{d M} + G}{2 ( MG ) ^{\frac{1}{2}}}

= 2 π \frac{\frac{3 \frac{d r}{d M} r ^{\frac{1}{2}}}{2} GM - \frac{M \frac{d G}{d M} + G}{2 ( MG ) ^{\frac{1}{2}}} r ^{\frac{3}{2}}}{( GM ) ^{2}}

化简

2 π \frac{\frac{3 \frac{d r}{d M} r ^{\frac{1}{2}}}{2} GM - \frac{M \frac{d G}{d M} + G}{2 ( MG ) ^{\frac{1}{2}}} r ^{\frac{3}{2}}}{( GM ) ^{2}} : \frac{π ( 3 M \frac{d r}{d M} G r ^{\frac{1}{2}} - ( M \frac{d G}{d M} + G ) r ^{\frac{3}{2}} )}{M ^{\frac{3}{2}} G ^{\frac{3}{2}}}

= \frac{π ( 3 M \frac{d r}{d M} G r ^{\frac{1}{2}} - ( M \frac{d G}{d M} + G ) r ^{\frac{3}{2}} )}{M ^{\frac{3}{2}} G ^{\frac{3}{2}}}

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