Originally Posted by
Jhevon note that $\displaystyle f(x)$ is defined by the limit definition of the derivative of $\displaystyle \csc x$
thus, let $\displaystyle g(x) = \csc x$
we have that $\displaystyle f(x) = g'(x)$
so $\displaystyle f'(x) = g''(x)$
therefore, $\displaystyle f' \left( \frac {\pi}4 \right) = g'' \left( \frac {\pi}4 \right) $
now find $\displaystyle g''(x)$ and you will have your answer
are you required to find the derivative by evaluating the limit? i doubt it. it will especially be a pain doing it for the second derivative