Calculus: Early Transcendentals, Chapter 3, 3.3, Section 3.3, Problem 15

To find
y'=d/(dx) x*e^x csc(x)
let a= x*e^x so,
y'=d/(dx) (a*csc(x))
= (d/(dx) a) *csc(x) + a* d/(dx)(csc(x))
substituting a = x*e^x so a' = (d/(dx) a) = (d/(dx) x*e^x) = x*e^x + e^x and
d/(dx)(csc(x)) = -csc(x)cot(x)
so ,
y'=(d/(dx) a) *csc(x) + a* d/(dx)(csc(x)) = (x*e^x + e^x)*csc(x) + (x*e^x)* (-csc(x)cot(x)) = (e^x)(1+x)*csc(x)- (x*e^x)* (csc(x))*(cot(x)) = (e^x)*csc(x)[(1+x)-x*(cot(x))]