I would do what you have suggested
Make and
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.
Thank you!
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'