A partial derivative is a type of differentiation. The
function we looked at earlier had one variable (x). A partial derivative, on
the other hand, applies when a function has two or more variables. For
instance, if a function is f(x, y), where the value of the function depends on
the input values 'x' and 'y', then differentiating with respect to just one
variable is called partial differentiation. For example, f(x, y) = 2x2
+ 3y + 4 is a function with two variables that can be partially differentiated.
Partial Differentiation
When differentiating with respect to 'x', 'y' is
considered constant and is unaffected by differentiation. The symbol ∂,
pronounced "partial," is placed in front of both the function and the
variable to indicate partial differentiation.
Application of Partial
Derivative (Cited from Wikipedia)
The function x2 + xy + y2,
consisting of variables (x, y), represents a three-dimensional surface.
Partially differentiating this function with respect to 'x' yields 2x + y. When
x = 1 and y = 1, the result is 3, which represents the rate of change at that
point. Since this is partial differentiation with respect to 'x', the value of
'y' is fixed (in this case, set to 1), and we observe how the rate of change
varies with changing 'x'.
