Vector/Matrix Calculus

Calculus has its own limits :)

Definition:
matrix calculus is a specialised notation for doing multivariable calculus. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities.
The six kinds of derivatives that we are going to discuss can be organised in a matrix as below.

Here matrix is used in general term. For ex vectors are also 1 dimensional matrix and scalar is a single element matrix.

Scalar By Scalar:
If y is defined as continuous and differentiable function of x for the range (x1, x2) such that.

And it’s equal to the slope of function y at point x=t. It can also be understood as rate of change y w.r.t to infinitely small change in x.
Ex:

Vector by Scalar:

Ex:

Matrix by Scalar:

Ex:

Scalar By Vector:
Let y is a scalar function of n independent variables such that

Then the derivative of y w.r.t to vector x is given by the vector equation

Ex:

Vector By Vector:

Ex:

Scalar By Matrix

Ex: