In calculus, a partial derivative measures how a function changes when only one of its variables is varied, holding all other variables constant. It is denoted by the symbol â (partial derivative symbol).
Notation
The partial derivative of a function with respect to x is written as: or
. Similarly, the partial derivative of with respect to y is written as: or .
First Order Partial Derivatives
Chain Rule for Partial Derivatives
Extreme Values
Extreme values of a function of two variables occur at points where the function reaches its maximum or minimum values. These can be further categorized into:
Local Extrema
- Local Maximum: A point where the function takes on a greater value than at nearby points. Mathematically, at point (x, y), f(x, y) is greater than or equal to f(a, b) for all (a, b) in some neighborhood around (x, y).
- Local Minimum: A point where the function takes on a lesser value than at nearby points. Mathematically, at point (x, y), f(x, y) is less than or equal to f(a, b) for all (a, b) in some neighborhood around (x, y).
Global Extrema
- Global Maximum: A point where the function takes on the greatest value over its entire domain.
- Global Minimum: A point where the function takes on the least value over its entire domain.
Saddle Points
A saddle point is a point on the surface of a function where the slopes (derivatives) in all directions are either zero or do not exist, but it is neither a maximum nor a minimum. At a saddle point, the function resembles the shape of a saddle, hence the name. Mathematically, at point (x, y), the function has critical points where both partial derivatives are zero but the concavity changes along different directions.
In summary, extreme values represent the maximum or minimum points of a function, while saddle points are critical points where the function neither reaches a maximum nor a minimum but has zero slope in multiple directions.