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.

Note that it's unnecessary to treat $dx$ and $dt$ as independent entities ("infinitesimals") when solving a differential equation like this.
One way to solve it (other than just guessing the solution) is this:
\begin{align}
& x'(t) = x(t) \\
\implies & x'(t) - x(t) = 0 \\
\implies & x'(t) e^{-t} - x(t) e^{-t} = 0 \qquad (\heartsuit)\\
\implies & x(t) e^{-t} = C \quad \text{for some } C \in \mathbb R \quad (\spadesuit)\\
\implies & x(t) = Ce^t.
\end{align}
To go from ($\heartsuit$) to ($\spadesuit$), we took antiderivatives of both sides, using the product rule in reverse.
This is a standard trick which can be used to solve any first order linear ODE.
