I would do what you have suggested
To find you will need to use the product rule again.
I'm not sure where to begin with finding the derivative of this function:
f(x) = x * e^x * cscx
I understand the product rule and how it applies to problems that have two functions, but how would one incorporate a third function?
I've using the product rule on the first two, x & e^x, and then used that derivative and putting it in as one of the two functions in the product rule, along with cscx, but this seems like a sloppy/invalid method.
Think of (uvw)'= ((uv)w)' and treat (uv) as a single function:
((uv)w)'= (uv)'w+ (uv)w'. Now use the product rule on (uv). (uv)'= u'v+ uv" so (uvw)'= (u'v+ uv')w+ (uv)w'= u'vw+ uv'w+ uvw'.
That extends very easily to any number of functions: (uvw...z)'= u'vw...z+ uv'w...z+ uvw'...z+ ...+ uvw...z'.
Another way to get that is to use "logarithmic differentiation". If f= uvw...z then ln(f)= ln(uvw...z)= ln(u)+ ln(v)+ ln(w)+ ...+ ln(z). (ln(f))'= (1/f)f'= u'/u+ v'/v+ w'/w+ ... z'/z. Multiplying on both sides by f= uvw...z gives f'= u'vw...z+ uv'w...z+ uvw'...z+...+ uvw...z'