Question

Find the derivative and simplify. y = 2csc^3(sqrtx) Help?!

Answer 1

`y'=-(3csc^3(sqrtx)cot(sqrtx))/sqrtx`

Explanation:

`y=2csc^3(sqrtx)`

If we let `v= sqrt x` and `u=cscv`, we can define `y=2u^3`

Then `dy/dx = dy/(du)xx(du)/(dv)xx(dv)/dx`

`dy/(du) = 6u^2 = 6csc^2v = 6csc^2(sqrtx)`

`(du)/(dv) = -cscvcotv = -csc(sqrtx)cot(sqrt x)`

`(dv)/dx = 1/(2sqrtx)`

`thereforedy/dx=y'=6csc^2(sqrtx)xx -csc(sqrtx)cot(sqrtx) xx 1/(2sqrtx) = -(3csc^3(sqrtx)cot(sqrtx))/sqrtx`