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Does this function have a derivative? $$f_p(x)=\sum_{k=1}^{\lfloor \frac{p}{4qx}+\frac{1}{2}\rfloor}x^2csc^{2}\left(j_kx\right) $$ How do I derive this function?

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    What are $p$, $q$, and $j_k$? – 2017-01-13
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    @MichaelMcGovern probably just arbitrary parameters. Considering the function is a sum dependent on the floor function, I am pretty sure the derivative is everywhere defined. As for where it is defined, I have no clue. – 2017-01-13

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There are two threats to it having a derivative. One is that the cosecant can be undefined, which happens when $j_kx=n\pi$ for some $x,n$. This is canceled at $x=0$ by the $x^2$ factor. The other is when $\frac p{4qx}+\frac 12$ is an integer as you have a term removed from the sum as $x$ increases through that point. Other than these points you can use the facts that sums, products, and compositions of differentiable functions are differentiable.

Aside from those points, you can just take the derivative term by term inside the sum. Alpha gives $$\frac d{dx}(x^2 \csc^2(j_k x)) = 2 x \csc^2(j_k x) - 2 j_k x^2 \cot(j_k x) \csc^2(j_k x)$$