Construct a single-valued function increasing arbitrarily quickly at a point $x=x_0$?

Question

Title says it all.

(How) can we construct a function (a regular function, perhaps?) whose first derivative is arbitrarily large at a point $x = x_0 < \infty$?

Answer 1

Let $f(x)=-(x-x_0) \, \ln |x-x_0|$, extended by continuity at $x_0$. Then, its derivative is $f'(x)=-\ln |x-x_0|-1$ and $\lim_{x \to x_0} f'(x)=+\infty$.

As mentioned in the comments, the primitive of any convergent improper integral would do the job.