I'm not entirely sure how to approach this.
Change is part of a businesses growth.
Change is part of a business' growth.
Which one is correct and why?
This means "the growth of a business." So you want the possessive form, not the plural. Therefore only the second choice is correct.
However, the second choice will look wrong to many people. It is true that some people use an apostrophe alone to form the possessive case of words ending in an s or z sound, but this is much less common than adding 's. In this case, "business' " is much less common than "business's."
The word "business's" is pronounced exactly the same way as "businesses."