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With all due respect to Mr. Israel, his answer is incomplete and, I must apologize writing this, misleading. Let $f: \mathbb{R}^n \to \mathbb{R}^m$ so that the Jacobian $f'(x)$ is a $n \times m$ matrix for every $x$. Assume that $f(x)$ takes $T$ arithmetic operations to evaluate. Fix $x \in \mathbb{R}^n$. Forward AD evaluates $f'(x)$ in $O(1)Tn$ arithmetic ...