In this video tutorial, next concept “Partial Derivative” will be explained.

The concept of simple derivative is explained in 1st video.

Derivative, simply, means change in function with respect to variable (say x).

In two dimensional plot, function f(x) is dependent on single variable x. So, derivative is calculated or denoted as df/dx.

If two variables are considered, function f(x,y) is dependent on two variables x and y. If the function is dependent on two variables or multiple variables, partial derivative needs to be calculated. Partial derivative is represented or denoted as as ∂f/∂x i.e. partial derivative wrt x and ∂f/∂y i.e. partial derivative wrt y.

Definition of Partial Derivative:

Partial Derivative of function of several variable is its derivative wrt one of its variables, with others held constant.

Example on simple derivative is explained. Example is f(x) = x3

Formula used for calculation of simple derivative is also mentioned.

One more example based on partial derivative is explained. Example is f(x,y) = x3 + y3 + 3xy

Above examples are single derivative. For single derivative, two combinations are possible.

Second derivative:

For second derivative, four combinations are possible.

The four possible combinations are:

Differentiating x twice, differentiating y twice, differentiation of x and y, differentiation of y and x.

For function of two variables, there will be four possible second order derivatives.

(fx)x simply means differentiating function with respect to x and again differentiating that result with x. Same applies for (fy)y.

Third combination (fx)y: differentiating x first and result differentiating with respect to y.

Last combination is (fy)x

Second derivative is applicable for 2 variables.

For n variables, hessian matrix is used.

Hessian matrix:

Hessian matrix is basically a square matrix of second order partial derivative. For n variables, hessian matrix is represented as explained in the tutorial. Also, representation for 2 variables is explained.

Example based on Hessian matrix is explained. Example is x^3 + y^3 + 3xy.

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