Verified answer

I will re-write it like this:

y = exp { u * [cos(u) + c*u]} = exp { u * cos(u) + c*u^2]}

Start by using the chain rule... take the derivative of the outside (exponential), which gives the same thing times the derivative of the inside, to do the derivative of the inside you need to use the product rule and then differentiate normally...

dy/du = exp { u * [cos(u) + c*u]} * { [cos(u) + u*sin(u)] + [2*c*u] }

Just to be perfectly clear, I will re-write as:

dy/du = { cos(u) + u*sin(u) + 2*c*u} * exp { u * [cos(u) + c*u]}

----------------------------

y = csc(x) (x + cot(x))

recall that d/dx { csc (x ) } = - csc(x) * cot(x)

and that d/dx {cot (x) } = - csc^2(x)

Use the product rule on the csc and the (x + cot):

dy/dx = [-csc(x) * cot(x)] * [x + cot(x)] + csc(x) * [1 - csc^2 (x) ]

You can simplify if you want... I don't care to since I think it's more effort than it's worth.