Basic partial derivative verification question

Question

I have F=y^'(1+x²y^'), and I need to find ∂F/∂y and ∂F/∂y'

The answer I get for ∂F/∂y is 0, which is the correct one. However for ∂F/∂y' I should get 1+2x²y', but for some reason I get 1+2x²y'+y^'x². Could someone please tell me where I went wrong?

My try: Using the product rule we get ∂F/∂y'=∂/∂y'(y')(1+2x²y^')+y'∂F/∂y'(1+x²y^')=1*(1+2x²y')+y'x².

Answer 1

F = y^'(1+x²y^')

= y' + x²y'²

∂F/∂y' -> treat x² as constant

= 1 + 2x²y'

∂F/∂y -> treat y' and x² as constant

= 0 + 0

= 0

Answer 2

It might help to rewrite your steps because it seems like you just miscopied one number. Since the pieces for your product rule are y' and 1+x^2y', the product rule says ∂F/∂y'=∂/∂y'(y')(1+x²y^')+y'∂F/∂y'(1+x²y^')=1*(1+x²y')+y'x^2, which simplifies to the answer you want.