= Solution
For the <force> magnitude in <Newton's law of universal gravitation>, the <total differential> is
$$
\boxed{dF=\frac{Gm_2}{r^2}\,dm_1+\frac{Gm_1}{r^2}\,dm_2-\frac{2Gm_1m_2}{r^3}\,dr.}
$$
Dividing by the positive value $F$ gives the particularly useful relative form
$$
\boxed{\frac{dF}{F}=\frac{dm_1}{m_1}+\frac{dm_2}{m_2}-2\frac{dr}{r}.}
$$
For finite small changes this gives the first-order approximation to the fractional change; higher-order products of increments are omitted.
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