The 4 out the front should be a 5. I assume this is a careless oversight on your part.
\(y=(sinu)^5\\ \frac{dy}{du}=5(sinu)^4\)
I have answered it myself and the rest of your answer is correct. (I only compared our end end results.)
I also double checked with Wolfram Alpha and it agrees too
https://www.wolframalpha.com/input/?i=y%3D%5Bsin(e%5E(t%5E2)-cos(t))%5D%5E5++differentiate
.