/**
 * Forward-mode automatic differentiation over the plain functions surface.
 *
 * The elementary typed functions (`add`/`subtract`/`multiply`/`divide`/`pow`/
 * `sin`/`cos`/`tan`/`exp`/`log`/`sqrt`/`square`/`cbrt`/`cube`/`abs`) carry `Dual`
 * signatures, so a function written with them differentiates exactly when
 * evaluated on a seed `Dual`:
 *
 *   import { multiply, sin, derivativeAt } from '@danielsimonjr/mathts-functions';
 *   derivativeAt((x) => multiply(sin(x), x), 2)   // sin(2) + 2·cos(2)
 *
 * This complements the symbolic `derivative` (which returns an expression) and
 * the autograd package's tensor `grad`: `derivativeAt` evaluates the exact
 * scalar derivative of an ordinary numeric function at a point, with no finite
 * differences and no manual tape.
 */
import { Dual } from '@danielsimonjr/mathts-core';
/** Function of one dual number, as the forward-mode derivative helpers take it. */
export type DualFn = (x: Dual) => Dual;
/** Exact derivative of `fn: ℝ → ℝ` at `x0` (forward-mode AD). */
export declare function derivativeAt(fn: DualFn, x0: number): number;
/** Both the value and the exact derivative of `fn` at `x0`. */
export declare function valueAndDerivativeAt(fn: DualFn, x0: number): {
    value: number;
    deriv: number;
};
/**
 * Gradient of `fn: ℝⁿ → ℝ` at `x` by n forward passes (seed one coordinate at a
 * time). `fn` receives an array of `Dual`s and must return a scalar `Dual`.
 */
export declare function gradientAt(fn: (x: Dual[]) => Dual, x: readonly number[]): number[];
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