differential

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 x₀, written as f′(x₀), is defined as the limit as Δx approaches 0 of the quotient Δy/Δx, in which Δy is f(x₀ + Δx) − f(x₀). Because the derivative is defined as the limit, the closer Δx is to 0, the closer will be the quotient to the derivative. Therefore, if Δx is small, then Δy ≈ f′(x₀)Δx (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 x₀, in this case the perfect square 16, results in a simple calculation of f′(x₀) as 1/8 and Δx as 1, giving an approximate value of 1/8 for Δy. 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.