Need explanation for simple differential equation I can't figure out this really simple linear equation:
$$x'=x$$
I know that the result should be an exponential function with $t$ in the exponent, but I can't really say why. I tried integrating both sides but it doesn't seem to work. I know this is shameful noob question, but I would be grateful for any hints.

$$x'=x \Rightarrow \frac{dx}{dt} \Rightarrow \frac{1}{x} dx = 1 \ dt \Rightarrow \int \frac{1}{x} dx = \int 1 \ dt \Rightarrow \ln(x)= t + C \Rightarrow x(t)=e^te^c.$$
$\cdot \ \text{Let A}=e^c \ \text{then} \ x(t) = Ae^t$
