Question

need help with the following attachment

Answer 1

I think the answer is cos(e^(t)) e^(t)+e^(sin t)sin

Answer 2

Looks like chain rule:
For sin(e^t), u=e^t and du=e^t
Substitute e^t with u and derive sin(u), then multiply by du: cos(u)*du
Replace u & du: cos(e^t)*e^t

For e^sin(t), u=sin(t) and du=cos(t)
Substitute sin(t) with u and derive e^u, then multiply by du: e^u*du
Replace u & du: e^sin(t)*cos(t)

So you're entire answer is:
\[\cos(e^t)*e^t + e^{\sin(t)}*\cos(t)\]

Answer 3

Thanks!