Question

已知x=t(1-cost),y=tcost,确定了y=f(x),求dy/dx和d^2y/dx^2,

Answer 1

已知x=t(1-cost),y=tcost,确定了y=f(x),求dy/dx和d²y/dx²

y'=dy/dx=(dy/dt)/(dx/dt)=(cost-tsint)/(1-cost+tsint)；

y''=d²y/dx²=dy'/dx=(dy'/dt)/(dx/dt)

={[(1-cost+tsint)(-sint-sint-tcost)-(cost-tsint)(sint+sint+tcost)]/(1-cost+tsint)²}/(1-cost+tsint)

=[(1-cost+tsint)(-2sint-tcost)-(cost-tsint)(2sint+tcost)]/(1-cost+tsint)³

=[-(1-cost+tsint)(2sint+tcost)-(cost-tsint)(2sint+tcost)]/(1-cost+tsint)³

=[(2sint+tcost)(1+cost-tsint-cost+tsint)]/(1-cost+tsint)³

=(2sint+tcost)/(1-cost+tsint)³