You can actually delete the calculation table as I have placed the entire formula in column D; the table is just there to make it easier to understand how the formula works.
As with almost any Excel problem, there are many different ways this could be done. I chose this approach because you mentioned that the tax rates are subject to change and that you didn't want to have to rewrite your formulas whenever there is a change in tax rates.
If the tax rates were never going to change, this could also be done by using IF functions or the IFS function. For example, here's an IF function that calculates the highest rate of tax:
As you can see, this checks whether the gross was above 3533 and then adds the basic rate of tax to 22% of the excess.
The IF function is covered in depth: Expert Skills Session 3 (Lesson 3-5: Use the IF logic function).
You could calculate the other tax rates as follows:
You could place these in separate columns or add them all together in a single formula, but you could also combine them together more efficiently using the IFS function like this:
The IFS function offers and efficient way to create logical formulas with multiple criteria like this one.
You can find a full explanation of IFS in: Expert Skills Session 3 (Lesson 3-28: Use the IFS function).
The logical formula approach makes for a tidier workbook without any additional tables, but it also makes it harder to understand and change the tax rates. The VLOOKUP approach makes tax rates easy to change, and new rates can even be added without needing to modify any formulas.