Relationship Between the Probability Current and Continuity Equation

[darfmore]

I'm currently reading through a textbook by David Miller and attempting to teach myself quantum mechanics to assist with my electrical engineering. I have run into a little trouble trying to understand how the probability current satisfies the continuity equation with a probability distribution as shown:

(The probability current equation that is defined in the textbook is given in the attached image)

(d/dt)P(x,t) + (d/dx)J(x,t) = 0, where P(x,t) = |ψ(x,t)|^2

This is an assumption made in deriving further applications about the probability current and the text suggests that I derive the relationship to practice the mathematics of quantum mechanics but I can't see how the expression is valid.
Any ideas on how to go about it? Thanks.

 

[Sir Beaver]

I would suggest by starting with the probability, write the probability as \psi^* \psi, and take the time derivative. Make connection with the Schrödinger equation after that.