Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).

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Interchanging pointwise limit and derivative of a sequence of C1 functions

Assume the following: $f_n$ is a sequence of C1 functions on $[0,1]$ $f_n(x) \rightarrow 0$ for pointwise. $f'_n(x) \rightarrow g(x)$ pointwise. Is it true that $g(x) = 0$ almost everywhere? I think ...
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Lyapunov function for non-autonomous non-linear differential equations

I have read some lecture notes about Lyapunov’s Second Method for autonomous system. Now, I want to deal with the stability of a non-autonomous system. Suppose there is a non-autonomous non-linear ...