Derivatives of 2 variable functions

A 2 variable function forms a plane with height defined at every point by the value of the function.
#todo add example of 3d view of from desmos

The derivative of the 2 variable function is composed of partial derivatives in the both the x and y axes.

Partial derivatives

The partial derivative in the x axis is the Instantaneous velocity of when moving in the x direction.
This derivative is typically denoted as
The same applies for the y axis which is denoted as:

How to calculate partial derivatives

When calculating a partial derivative of the function in the axis x we simply treat all the other variables () as constants (as if they were a number like 1, 2 or 3). The sample applies for the y axis.
Example:

The function gradient

The gradient of the function is defined as:

It is a vector field where at every point it is equal to a vector compose of the partial derivatives in he x and y axis.

When taking a vector we can calculate the rate of change of of the original function in the direction of at point using the dot product between the fucking gradient and the fucking normalized vector :

Tangent plane

To calculate the tangent plate to a function at point we can use the following formula:

Double derivatives

You can also derive the partial derivatives of a partial derivative in both y and x axis.

For the example function the results might look something like this:

Hessian matrix

A hessian matrix is a matrix composed of the partial derivatives of a function that describes its curvature. The matrix takes the following form: