DemoPackage__index.md

Dual Number Polynomial Differentiation

DualNumber

Polynomial

Differentiation

By using dual numbers with polynomials we can evaluate the derivative! Consider a polynomial P(x) = p_0 + p_1 x + \cdots + p_n x^n. Then,

$$\begin{aligned} P(a + bϵ) &= p_0 + p_1 (a + bϵ) + \cdots + p_n (a + bϵ)^n \\\ &= p_0 + p_1 a + p_2 a^2 + \dots \\\ &+ p_1bϵ + 2p_2 abϵ + 3p_3 a^2 b ϵ + \cdots \\\ &= P(a) + bP^\prime(a)ϵ \end{aligned}$$

derivative