Is there a differentiable function such that $f(\mathbb Q) \subseteq \mathbb Q$ but $f'(\mathbb Q) \not \subseteq \mathbb Q$?
Nice question.
Here is my solution. It would be nice to know if there is a simpler example.
This is basically a "spline" example. First, if $A,B,C,D,r,s$ are all rational numbers, and $r < s$, then there exist rational numbers $a,b,c,d$ so that the cubic polynomial $f(x) = ax^3+bx^2+cx+d$ satisfies $f(r)=A,f'(r)=B, f(s)=C, f'(s)=D$. The proof is: write down the system of equations and solve for $a,b,c,d$. As long as $r \ne s$ the solution is rational in $A,B,C,D,r,s$.
Remark 1. Computation shows that if $B=D=0$, then the graph of $f$ on $[r,s]$ lies inside the rectangle with opposite corners $(r,A)$ and $(s,C)$. [Differentiate the cubic, then factor to find that the derivative vanishes at the two endpoints $r,s$.]
The construction First, choose your favorite irrational number, let's say $\pi$. Consider the two curves $\phi_1(x)=\pi x$ and $\phi_2(x) = \pi x + x^3$.
Of course, any function $f$ with graph between these two has $f'(0)=\pi$. Define such a function by using a sequence of rational "knots" $r_n \searrow 0$ and $A_n$ so that $$ \phi_1(r_n) < A_n < \phi_2(r_n) $$ Then fill in with the splines as explained, to get a function $f$ with $f(r_n) = A_n$ and $f'(r_n) = 0$ for all $n$. Do the same thing on the negative side. Finallly let $f(0)=0$.
This gives us $f : \mathbb R \to \mathbb R$ such that $f(\mathbb Q) \subseteq \mathbb Q$, $f$ is differentiable except possibly at $0$, and $$ \lim_n \frac{f(r_n)-f(0)}{r_n-0} = \pi $$ is irrational.
What remains: we need to choose the sequences $r_n$ and $A_n$ so that the rectangles with opposite vertices $(r_n,A_n)$ and $(r_{n+1},A_{n+1})$ lie entirely between $\phi_1$ and $\phi_2$. Then by Remark 1, $f$ remains between $\phi_1$ and $\phi_2$, so $f'(0)=\pi$.
This answer shares the use of cubic splines with that of GEdgar. Consider the following polynomial function: $$ p(x)=2(x-1)^3-3(x-1)^2+1$$ $p$ satisfies $p'(1)=p'(2)=0$. Studying $p$ in the interval $[1,2]$ we see that in this interval $p$ is non negative and strictly decreasing, with $p(1)=1$ its maximum and $p(2)=0$ its minimum. Therefore the function: $$ h(x)= \begin{cases} 1& |x|\leq 1\\ p(|x|)& 1<|x| \leq 2\\ 0 &|x|>2 \end{cases} $$ is in $C^1(\mathbb{R})$. Now set: $$ g_n(x)=h(nx)/n!$$ and consider $g(x):=\sum_{n=0}^{+\infty} g_n(x)$.
What we are going to prove is that $f(x):=xg(x)$ is a differentiable function and satisfies $f(\mathbb{Q})\subseteq \mathbb{Q}$ and $f'(0)=e$.
Let's first discuss the values taken by $f$ on the rationals. The function $g_n$ takes only rational values in every rational point and for $n>0$ we have $g_n$ is zero outside $[-2/n,2/n]$. This means that for every $q \in \mathbb{Q}-\{0\}$ the sequence $(g_n(q))$ is eventually zero, so that $g(q)$ is in fact a finite sum of rationals and $f(q)=qg(q)$ is also rational. We also have $g(0)=e$ but clearly $f(0)=0\cdot e=0 \in \mathbb{Q}$.
To show $f$ is differentiable, it is enough to show $g$ is. Since $g'_n(x)= h'(nx)/(n-1)!$ for $n>1$, we have $\sum g'_n$ converges uniformly on $\mathbb{R}$ and since $\sum g_n(0)$ converges we deduce that also $g$ is differentiable, with $g'=\sum g_n'$.
Finally, $f'(0)= g(0)=e$. This concludes the proof.
Here is another answer. Define $h:[0,+\infty)\to \mathbb{R}$ as follows: $$h(x):= \frac{1}{(n+1)!}(n+1-n(x-n)) \quad \text{for } x\in [n,n+1)$$ The function $h$ is continuous and satisfies $h(k)=1/k!$ for every $k$ integer in its domain. Moreover: $$ \int _0^{q}h(t)dt \in \mathbb{Q} \quad \forall q \in \mathbb{Q}$$ since on every interval of the type $[n,n+1)$ the function $h$ is an affine function with rational coefficients. A direct computation shows: $$ \int _0^{\infty}h(t)dt = \sum_{k=0}^{+\infty} \frac{1}{2}\big( \frac{1}{k!} + \frac{1}{(k+1)!}\big) = e - \frac{1}{2}$$ Define now $f:\mathbb{R}\to \mathbb{R}$ as follows: $$f(x):= \begin{cases} 0 & x=0\\ x \int _0^{1/|x|}h(t)dt & x\neq 0 \end{cases} $$ Then $f$ satisfies $f(\mathbb{Q})\subseteq \mathbb{Q}$ and it is differentiable in $x\neq 0$ for the fundamental theorem of integral calculus. What about $x=0$? We have: $$\lim_{x\to 0}\frac{f(x)-f(0)}{x-0} = \lim_{x\to 0}\frac{x \int _0^{1/|x|}h(t)dt}{x}= \int _0^{+\infty}h(t)dt = e - \frac{1}{2}$$ Hence $f$ is differentiable on the whole real line and $f'(0)= e-1/2 \notin \mathbb{Q}$