I am not sure I understood your question about the importance of the Mathematics branches you mentioned on our life. You can say, It is every thing in our life, since any thing on our life and on the the world generally is transferred by either a direct or indirect way to these branches to be solved and clarifying the situation based on the mathematical solution results.
In PhD entrance interview one of my friend faced the question is what is scientific answer to use foirier transformation and Laplace's also? That's y i got doubt the remaining differentials, complex analysis, Matrices also. Why these are all...what are their specifications to use in mathematics. I have studied formulas only to solve them but i didn't get idea why we have to use. Why we need these are all?