Here's an attempt to salvage Matthew Pancia's solution, which unfortunately depended on an uncountable union over all possible derivatives.
Given $f$ we can define, in the obvious way, a continuous function $F:\mathbb R\times(\mathbb R\setminus 0)\to \mathbb R$ such that $f$ is differentiable at $x$ exactly when $\lim_{h\to 0} F(x,h)$ exists. The usual formalization of this is $\exists y:\forall\varepsilon:\exists \delta:\forall h: |h|<\delta\Rightarrow |F(x,h)-y|<\varepsilon$ Classically all of the variables here are real, but it is easy to see that we can restrict $\varepsilon$ and $\delta$ to $\mathbb Q$ without changing the meaning. We can also restrict $h$ to $\mathbb Q$ because $F$ is continuous. However, it is essential that $y$ can be an arbitrary real, because otherwise we would be looking for points where $f$ is differentiable with rational derivative, which is something quite different.
However, we can also formalize the existence of a limit like $\forall\varepsilon:\exists \delta:\exists Y:\forall h: |h|<\delta \Rightarrow |F(x,h)-Y|<\varepsilon$ This works because $\mathbb R$ is complete; it is essentially the same as changing "has a limit" about a sequence to "is Cauchy". The arguments that $\varepsilon$, $\delta$ and $h$ can be restricted to the rationals work as before, but now $Y$ can also be taken to be a rational in each case.
For each particular choice of $\varepsilon$, $\delta$, $Y$, and $h$, the set of $x$ such that $|h|<\delta\Rightarrow |F(x,h)-Y|<\varepsilon$ is open and therefore Borel.
Now handle each of the quantifiers from the inside out: For each choice of $\varepsilon$, $\delta$, and $Y$, the set of $x$ such that $\forall h: |h|<\delta \Rightarrow |F(x,h)-Y|<\varepsilon$ is a countable intersection of Borel sets and therefore Borel. For each choice of $\varepsilon$ and $\delta$ the set of $x$ such that $\exists Y:\forall h: |h|<\delta \Rightarrow |F(x,h)-Y|<\varepsilon$ is a countable union of Borel sets and therefore Borel. And so forth. At the top we find that the set of points of differentiability is Borel and thus in particular measurable.