> [!NOTE] Lemma (The derivative of the exponential) > The [[Fréchet Differentiation|derivative]] of the [[Real Exponential Function|exponential]] is given by $\exp'(x)=\exp(x)$ *Proof*. Note that $\exp(x) = \sum_{n=0}^{\infty} \frac{x^{n}}{n!}$ By [[Power Series is Termwise Differentiable within Radius of Convergence|derivative of power series]], $\exp'(x) = \sum_{n=0}^{\infty} \frac{n}{n!}x^{n-1} = \sum_{n=1}^{\infty} \frac{x^{n-1}}{(n-1)!}=\exp(x)$