Thank you to Eddie from Wilson's School for submitting this very clear solution:
The simplest polynomial in $x$ is a linear one, such as $3x - 5$, and its
derivative is 3. Finding the derivative of a polynomial is simple: multiply
each term by its power in $x$, and reduce the power by 1. Since the power in
$x$ of every term is being reduced by 1, the derivative of the function will
never be equal to the function... for a finite-order function. If there is
no highest order term, i.e. the order is infinite, then it is possible.
For $p(x) = p'(x)$ to be true, each term $a_n x^n$ in $p(x)$ (where $a_n$ means "a
subscript n" and $a_n$ is the coefficient of $x^n$) must be equal to the
derivative of the next term $a_{n+1}x^{n+1}$ in p(x), which is
$(n+1) a_{n+1} x^n$. So:
So the coefficients of $x^n$ follow a sequence, such that the next
coefficient (that of $x^{n+1}$) is equal to the previous coefficient divided
by $n+1$. If we let $a_1 = 1$: