Parallel 1st Order Reaction Rates

Consider some general parallel first order process, where $\ce{A -> B}$ and $\ce{A -> C}$ (rate constants $k\_1$ and $k\_2$ respectively)


The overall rate equation of this process is said to be $-\frac{\mathrm d[A]}{\mathrm dt} = k\_1[\mathrm A] + k\_2[\mathrm A]$


I get the impression that this is derived from the addition of each individual rate equation, i.e. the addition of $-\frac{\mathrm d[\mathrm A]}{\mathrm dt} = k\_1[\mathrm A]$ and $-\frac{\mathrm d[\mathrm A]}{\mathrm dt} = k\_2[\mathrm A]$ would result in the overall rate equation shown above.


However, if this is the case, then why isn’t the rate equation actually $-\frac{2\,\mathrm d[\mathrm A]}{\mathrm dt} = k\_1[\mathrm A] + k\_2[\mathrm A]$? I mean, you’re adding two $-\frac{\mathrm d[\mathrm A]}{\mathrm dt}$ terms together; and so considering basic algebra, surely this would result in $-\frac{2\,\mathrm d[\mathrm A]}{\mathrm dt}$?


Or do differentials just not work in this way, and I’m simply confusing myself?




There is no need adding the number $2$ in front of the definition of the rate. So the rate equation is :
$$-d[A]/dt = k\_1 [A] + k\_2[A] = (k\_1 + k\_2)[A]$$


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