First-Order Reaction/Proof 1

Proof
From the definition of a first-order reaction, the rate of change of the quantity of the substance is proportional to the quantity of the substance present at any time.

As the rate of change is a decrease, this rate will be negative.

Thus the differential equation governing this reaction is given by:
 * $- \dfrac {\mathrm d x} {\mathrm d t} = k x$

for $k \in \R_{>0}$.

At time $t = 0$ we have that $x = x_0$.

So:
 * $x_0 = e^c e^0 = e^c$

and hence the result
 * $x = x_0 e^{-k t}$