Higher Order Partial Derivatives - Clairaut's Theorem - Calculus

In this video I will show you what is Clairaut's Theorem and how to verify it by some examples.
In mathematics, partial derivative of a function of several variables is its derivative with respect to one of those variables while the other variables are held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.
The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it for partial differences.
Applications:
Optimization: Partial derivatives are used in finding critical points and optimizing multivariable functions.
Physics: They describe rates of change in physical systems, like temperature gradients or velocity fields.
Engineering: Used in modeling systems with multiple input parameters.
Economics: Measure sensitivity of outcomes (e.g., profit, demand) to changes in variables like price or cost.