= Solution
View $a=a(b,c)$ with $b=b(d,e)$ and $c=c(d,e)$. Hold $e$ fixed and apply the <multivariable chain rule>:
$$
da=\left(\frac{\partial a}{\partial b}\right)_c db
+\left(\frac{\partial a}{\partial c}\right)_b dc.
$$
Dividing the changes along the fixed-$e$ variation by $dd$ gives exactly the two terms in the question. Thus the simpler expression is
$$
\boxed{\left(\frac{\partial a}{\partial d}\right)_e.}
$$
The subscripts specify which coordinate is held fixed and must be retained.
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