Differential | Calculus, Equations, Solutions | Britannica (original) (raw)

differential, in mathematics, an expression based on the derivative of a function, useful for approximating certain values of the function. The derivative of a function at the point x_0, written as f_′(x_0), is defined as the limit as Δ_x approaches 0 of the quotient Δ_y/Δ_x, in which Δ_y_ is f(x_0 + Δ_x) − f(x_0). Because the derivative is defined as the limit, the closer Δ_x is to 0, the closer will be the quotient to the derivative. Therefore, if Δ_x_ is small, then Δ_y_ ≈ f_′(x_0)Δ_x (the wavy lines mean “is approximately equal to”). For example, to approximate f(17) for f(x) = Square root of√_x, first note that its derivative _f_′(x) is equal to (_x_−1/2)/2. Choosing a computationally convenient value for _x_0, in this case the perfect square 16, results in a simple calculation of f_′(x_0) as 1/8 and Δ_x as 1, giving an approximate value of 1/8 for Δ_y. Because f(16) is 4, it follows that f(17), or Square root of√17, is approximately 4.125, the actual value being 4.123 to three decimal places.

This article was most recently revised and updated by William L. Hosch.