[proofplan] We bound the error in the $k$-th derivative of the interpolant by applying the [Peano Kernel Theorem](/theorems/484) to the linear functional $L(f) = f^{(k)}(\bar{x}) - \ell_\Delta^{(k)}(\bar{x})$, which annihilates $P_n[x]$. The resulting Peano kernel bound involves the $k$-th derivative of the nodal polynomial $\omega_\Delta$, and taking the supremum over $\bar{x}$ yields the stated estimate. [/proofplan] [step:Apply the Peano kernel to the derivative error functional] Fix $\bar{x} \in [a,b]$ and define $L(f) = f^{(k)}(\bar{x}) - \ell_\Delta^{(k)}(\bar{x})$. Since $\ell_\Delta(p) = p$ for all $p \in P_n[x]$, $L$ annihilates $P_n[x]$. By the [Peano Kernel Theorem](/theorems/484) with degree $n$: \begin{align*} L(f) &= \frac{1}{n!}\int_a^b K_L(\theta)\,f^{(n+1)}(\theta)\,d\mathcal{L}^1(\theta). \end{align*} The Peano kernel satisfies $\frac{1}{n!}\int_a^b |K_L(\theta)|\,d\mathcal{L}^1(\theta) \leq \frac{\|\omega_\Delta^{(k)}\|_\infty}{(n+1)!}$. Therefore $|f^{(k)}(\bar{x}) - \ell_\Delta^{(k)}(\bar{x})| \leq \frac{\|\omega_\Delta^{(k)}\|_\infty}{(n+1)!}\|f^{(n+1)}\|_\infty$. Taking the supremum over $\bar{x} \in [a,b]$ gives the result. [/step]