Database Performance

A software development team is working on improving the performance of a database query optimisation tool. The tool is designed to analyse and optimise SQL queries to improve database performance for a web application. The team needs to understand the concept of slope of a function at a point to identify critical areas where query performance can be enhanced. The team uses the function, $f(x)=\frac{x+8}{(3x+1{)}^2}$ to help them determine the rate of change of query execution time.

a$.$ Calculate the rate of change of the function mentioned in the scenario above at the point $(0,8).$

$(5$ Marks$)$

b$.$ What are some potential challenges or limitations associated with using the concept of slope of a function at a point to optimize database query performance?

$(3$ Marks$)$