`f(x) = x e^x csc(x)` Differentiate.

Textbook Question

Chapter 3, 3.3 - Problem 15 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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kspcr111 | In Training Educator

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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))]`

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