= Solution
Take the <total differential> of the <logarithm> of the <relativistic energy>, with the <speed of light> fixed:
$$
\ln E=\ln m+2\ln c-\frac12\ln\left(1-\frac{v^2}{c^2}\right).
$$
Using the <chain rule>,
$$
\frac{dE}{E}=\frac{dm}{m}+\frac{v\,dv}{c^2-v^2}
=\boxed{\frac{dm}{m}+\frac{v^2}{c^2-v^2}\frac{dv}{v}.}
$$
Every term is a dimensionless fractional change. The differential relation is exact; applying it to finite small changes is a first-order <Taylor expansion>.
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