Can I exchange limit and differentiation for a sequence of smooth functions?

real-analysis sequences-and-series analysis limits derivatives

Let $(f_n)_{n\in \mathbb N}$ be a sequence of smooth functions converging to some $f$.

Under what circumstances can I exchange limit and derivative?, i.e.

$$\lim_{n\rightarrow \infty} \frac{\partial f_n(x)}{\partial x} = \frac{\partial f(x)}{\partial x}$$


If you have a sequence of functions $(f_n)_{n\in \mathbb N}$ that are differentiable, and converge pointwise for some point $x_o$, and if their derivatives converge uniformly, say on a given interval [a,b], supposing they are real valued functions, then the sequence of functions $(f_n)_{n\in \mathbb N}$ is uniformly convergent to $f$ and what more, $$\lim_{n\rightarrow \infty} \frac{\partial f_n(x)}{\partial x} = \frac{\partial f(x)}{\partial x}$$

This is a standard theorem in Analysis. See Walter Rudin's Principle of Mathematical Analysis, 3rd Edition, Theorem $7.17$ for detailed proof.

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