How do we derive the expression for $e^x$ from $e$? Trying to go in historical order here and begin with Bernoulli's formulation for $e$:
$$e = \lim_{n \to \infty} (1 + 1/n)^n$$
How do we then make the jump to
$$e^x = \lim_{n \to \infty} (1 + x/n)^n$$
I had tried doing this:
$$e^x = (\lim_{n \to \infty} (1 + 1/n)^n)^x$$
$$e^x = \lim_{n \to \infty} (1 + 1/n)^{nx}$$
Let $m = nx$ so $n = m/x$. As $n$ goes to infinity, $m$ also goes to infinity, so:
$$e^x = \lim_{m \to \infty} (1 + x/m)^{m}$$
(although we could relabel with $n=m$ I just use $m$ to use a different one)
But I was told that I'm skipping many unproven assumptions doing this. Is there an easy way to prove what I am missing or is there an easier way to arrive at the result?

In what follows $n$ denotes a positive integer. This is a typical / conventional use of symbol $n$.
Using a little bit of algebra you can show that $$\lim_{n\to\infty} \left(1+\frac{x}{n}\right)^{n}=e^{x}\tag{1}$$ where $x$ is rational and $e$ is defined by $$e=\lim_{n\to\infty} \left(1+\frac{1}{n}\right)^{n}\tag{2}$$ Let's first assume that $x$ is a positive integer and then $$\left(1+\frac{x}{n}\right)^{n}=\left(\frac{n+x} {n} \right) ^n=\prod_{k=1}^{x}\left(\frac{n+k}{n+k-1}\right)^n\\=\prod_{k=1}^{x}\left(1+\frac{1}{n+k-1}\right)^{n+k-1}\left(1+\frac{1}{n+k-1}\right)^{1-k}$$ The first factor in the product on right tends to $e$ and second factor tends to $1$ so that the desired limit is $\prod_{k=1}^{x}e=e^x$. If $x=-y$ is a negative integer then we have $$\left(1+\frac{x}{n}\right)^{n}=\prod_{k=1}^{y}\dfrac{1}{\left(1+\dfrac{1}{n-k}\right)^{n-k}\left(1+\dfrac{1}{n-k}\right)^{k}}$$ and thus the desired limit is $1/e^{y}=e^x$.
For $x=0$ the result is obvious and thus we have established $(1)$ when $x$ is an integer. If $x$ is a rational number say $p/q$ where $p$ is an integer and $q$ is a positive integer then $$\left(1+\frac{x}{n}\right)^{n}=\sqrt[q]{\left(1+\frac{p}{qn}\right)^{qn}}$$ which tends to $\sqrt[q]{e^{p}} =e^x$ and our proof is complete. This step involves the continuity of function $f(t) =t^{1/q}$ and should be available to you already (this can also be proved using some algebraic manipulation).
For irrational $x$ we need a definition of irrational exponents and the proof depends on the chosen definition. The result holds even when $x$ is a complex number but the proof requires somewhat different tools (but again limited to basic algebraic manipulation).
