u=sin^-1(x+y) ,x=3t, y=4t^3

Question

solve by total derivative (chain rule) method

Answer 1

sin(u)=x+y, cos(u)du/dt=dx/dt+dy/dt.

dx/dt=3, dy/dt=12t², cos(u)du/dt=3+12t².

cos(u)=√(1-(x+y)²)=√(1-(3t+4t³)²), du/dt=3(1+4t²)/√(1-(3t+4t³)²).

1-(3t+4t³)²=1-t²(3+4t²)²=1-t²(9+24t²+16t⁴), so du/dt=3(1+4t²)/√(1-9t²-24t⁴-16t⁶).