#d/dx (csc x)#
# = d/dx (1/(sinx))#
Using the limit definition of the derivative:
Let #f(x) = cscx = 1/sinx#
# = lim_(h->0) frac{f(x+h)-h(x)}{h}#
# = lim_(h->0) frac{csc(x+h) - cscx}{h}#
# = lim_(h->0) frac{(1/sin(x+h) - 1/sinx)}{h}#
= #lim_(h->0) (sinx-sin(x+h))/(hsin(x+h)sinx)#
= #lim_(h->0) (2cos((x+x+h)/2)sin((x-x-h)/2))/(hsin(x+h)sinx)#
= #lim_(h->0) (-2cos(x+h/2)sin(h/2))/(hsin(x+h)sinx)#
= #lim_(h->0) (-2cos(x+h/2))/(hsin(x+h)sinx)xxsin(h/2)/(h/2)xxh/2#
= #lim_(h->0) (-cos(x+h/2))/(sin(x+h)sinx)xxsin(h/2)/(h/2)#
= #lim_(h->0) (-cos(x+h/2))/(sin(x+h)sinx)xxlim_(h->0)sin(h/2)/(h/2)#
= #-cosx/(sinxsinx)xx1#
= #-cotxcscx#
