DIFFERENTIAL

in mathematics, an expression based on the derivative (q.v.) of a function, useful for approximating certain values of the function. The derivative of a function at the point x0, written as f(x0), is defined as the limit as Dx approaches 0 of the quotient Dy/Dx, in which Dy is f(x0 + Dx) - f(x0). Because the derivative is defined as the limit, the closer Dx is to 0, the closer will be the quotient to the derivative. Therefore, if Dx is small, then Dy f(x0)Dx (the wavy lines mean is approximately equal to). For example, to calculate , the function involved is f(x) = , which has a derivative equal to 1/2x -1/2. If x0 is 16, then f(x0) is 1/8 and Dx is 1, giving an approximate value of 1/8 for Dy. Because f(16) is 4, it follows that f(17), or , is approximately 4.125, the actual value being 4.123 to three decimals.