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The question of interchanging differentiation and integration has been addressed on this platform (here) with sufficient references and answers.

I wanted to see with examples, the failure of this interchanging, to understand the importance of hypothesis in the theorem.

Q. Can one state simple examples of functions of two variables defined on a rectangle (or suitable other domain) where integration and differentiation are not interchangeable in Leibnitz rule? i.e. $$ \frac{d}{dx} \int_a^b f(x,t)dt \neq \int_a^b f_x(x,t)dt? $$

Beginner
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  • $a,b$ are constants. I am considering $f(x,t)$ to be defined on some rectangle $[c,d]\times [a,b]$. However, for the purpose of examples, one may take unbounded domain if necessary, but I wanted simple examples for failure – Beginner Aug 07 '19 at 07:26
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    See also this post. – Peter Foreman Aug 07 '19 at 07:32

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