I stumbled upon this questions while studying about complexities. So how should I go about finding a growth rate between two given complexities, say
Between $$O(\sqrt n ) \quad \text{and} \quad O(n)$$ For this I think the answer is $$f(n)= n^{\frac{3}{4}}$$
or
Between $$ O(n\cdot log(n)) \quad \text{and} \quad O(n^{2})$$
Your $f(n)$ is correct.
For the other question you can apply the same trick, so $f(n)=n\sqrt{n}\log(n)$.
See this link for more knowledge: https://en.wikipedia.org/wiki/Time_complexity#Table_of_common_time_complexities