An infinitesimal change in a variable or an infinitesimal change in a function resulting from a small change in the variable(s).

For example, if f(x) = 2x, then a change in f(x), symbol trianglef, resulting from a small change in x, symbol trianglex, would be symbol trianglef = f(x+symbol trianglex) - f(x) = 2(x+symbol trianglex) - 2x = 2symbol trianglex. The differential df is defined as the limit of symbol trianglef as symbol trianglex approaches infinitely small. In the above example, the differential is also called total differential because it takes into account changes in all of the variables (just one in this case). The derivative df/dx can be thought of as a ratio of two differential changes.

For a function with more than one variable, such as f(x,y) = 2xy, the change in f(x,y) resulting from a change in x, symbol trianglex, while keeping y constant, is called the partial differential:

symbol spacef(x+symbol trianglex,y) - f(x,y) = 2(x+symbol trianglex)y - 2xy = 2symbol trianglex y

The rate of change of f(x,y) with respect to x only is called the partial derivative.

Related Term: derivative

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