differentiaaliyhtälöperusteiset

Differentiaali, in mathematics often called the differential, refers to the linear part of the change of a function in response to a small change in its input. In single-variable calculus, for a differentiable function y = f(x), the differential dy is defined by dy = f′(x) dx, where dx represents a small change in x and f′(x) is the derivative. The ratio dy/dx equals f′(x) as the limit of Δy/Δx when Δx → 0; dy is the approximate change in y for a small Δx.

In multivariable calculus, for a function f: R^n → R, the differential at a point x is the linear map df_x(h) = ∑ ∂f/∂x_i (x) h_i, which can be written as df_x(h) = ∇f(x) · h. When x is changed by dx = (dx_1, ..., dx_n), the corresponding approximate change is dy ≈ df_x(dx) = ∑ ∂f/∂x_i dx_i. The differential is often described as the differential 1-form df = ∑ ∂f/∂x_i dx_i.

Example: for y = x^2, dy = 2x dx, so a small change dx in x yields an approximate change dy in y equal to 2x dx.

The differential underpins linear approximation and Taylor expansion, and it plays a central role in differential geometry through differential forms and in multivariable mappings via the Jacobian. In standard calculus, dy is an infinitesimal notion representing an approximate change; in rigorous treatments it is interpreted through linear approximation or nonstandard analysis.