Welcome and Thanks for requesting me.
There are 6 problems in all right? Need only answers? Let me know.
I need those questions answer , if you have step for each problem that is better. Thank you. Just four questions.
(1) f'(x) = (x^3 + 2x) e^x + (e^x)(3x^2 + 2)
= (e^x) (x^3 + 3x^2 + 2x + 2)
(2) y = (e^(ax^3)) a(3x^2)
= (3ax^2) e^(ax^3)
(3) y' = (e^-2t)(-4 sin 4t) + (cos 4t)(-2 e^-2t)
= (e^-2t)(-4 sin 4t - 2 cos 4t)
(4) f'(t) = (cos e^t) e^t + (e^(sin t)) cos t
(5) y' = [(e^u + e^-u)(e^u + e^-u) - (e^u - e^-u)(e^u - e^-u)] / (e^u + e^-u)^2
= 4 /(e^u + e^-u)^2
(6) y' = [1/2 √(1 + xe^-2x)] [x(-2e^-2x) + (e^-2x)]
= (e^-2x)(1 - 2x) / [2 √(1 + xe^-2x)]
I hope that helps. Let me know if you need any clarifications on my answers. I will be happy to provide them. Else, please give a rating for my answer.