can someone explain to me why the derivative and second derivative of this position vector r=ax+bty is r'=ax' +by'+ bty' ( why is there a by'...?)

Question

and r''=ax'' + cy' +cy' +cty'' ( why is there 2 cy' 's?

Answer 1

the product rule. Both t and y are functions of t, so

d/dt (bty) = b(y dt/dt + t dy/dt)
= b(y + ty')