Is My Chain Rule Derivation Correct for u*sin(x^2)?

[goldfronts1]

I am trying to find the first and second derivative using the chain rule of the following:

u sin(x^2)

This is what I have but I don't think it is correct. Can someone pls let me know?

first derivative: u * 2x cos(x^2) + sin(x^2) u'

second derivative:
u * 2( x * -2sin(x^2) + cos(x^2)) + 2xcos(x^2)* u' + sin(x^2)*u" + u'* 2xcos(x^2)

Any help thanks

 

[math20]

If u is a function of x and you are differentiating with respect to x, then it looks correct to me.