what is principal component analysis? What is the procedure for calculation of the same?

Answer 1

It's a linear transformation of the data that twiggs the data so that they get the following properties:
- The first coordinate captures the biggest possible fraction of the variance of the data
- The second coordinate captures the biggest possible fraction of the remaining variance
- etc

This is usefull if you have data with an awkwardly high number of coordinates. After PCA, you can pick the three or four (or whatever you want) first coordinates only and you have a smaller dataset containing most of the information in your original data.

It is done by computing the eigenvectors of the covariance matrix

Answer 2

Analysis, especially mathematical analysis is based on the consideration of behaviour of functions,etc.
So we use derivatives, simply put calculus is key to analysis.
You probably know the definition as the change of the dependant variable with the independant one.