Question

设函数z.f(x,y,z)=e的x+y²+3z,z=x²+2y,求∂z/∂x和∂z/∂y

Answer 1

问：设函数z.f(x,y,z)=e的x+y²+3z,z=x²+2y,求∂z/∂x和∂z/∂y

答：根据链式法则，可以将 ∂z/∂x 和 ∂z/∂y 表示为：∂z/∂x = ∂z/∂x + ∂z/∂z · ∂f/∂x= 2x e^(x+y²+3z) + ∂f/∂x= 2x e^(x+y²+3(x²+2y)) + e^(x+y²+3(x²+2y))= e^(x+y²+3(x²+2y)) · (2x+1)以及：∂z/∂y = ∂z/∂y + ∂z/∂z · ∂f/∂y= 4y e^(x+y²+3z) + ∂f/∂y= 4y e^(x+y²+3(x²+2y)) + 2e^(x+y²+3(x²+2y))= 2e^(x+y²+3(x²+2y)) · (2y+1)因此，∂z/∂x = e^(x+y²+3(x²+2y)) · (2x+1)，∂z/∂y = 2e^(x+y²+3(x²+2y)) · (2y+1)