Function of $x$ which diverges for $x\to c$ while derivative converges to a constant

As pointed out in the comments, the Mean Value Theorem shows why no such function exists.

We can pick any two points $a,b \in \rm{Dom}(f)$ such that $a,b \neq c$ and calculate the secant slope $S(a,b):= \frac{f(b)-f(a)}{b-a}$ formed by those two points $(a,f(a)), (b,f(b))$.

The Mean Value Theorem tells us $ \exists\delta \in (a,b): f'(\delta) = S(a,b)$.

However, since $\lim_{x\to c} f(x)= \infty$ then we can make $|S(a,b)|$ arbitrarily large by letting $b$ get arbitrarily close to $c$:

$$\forall a,b\neq c$$ $$\lim_{b \to c} |S(a,b)| = \infty$$

Therefore, $\lim_{x\to c} |f'(x)| =\infty$ since the sequence of derivatives are unbounded as you approach $c$.