homework and exercises - Calculating the Probability Current of a Travelling Wave

Calculate the probability current density vector $\vec{j}$ for the wave function : $$\psi = Ae^{-i(wt-kx)}.$$

From my very poor and beginner's understanding of probability density current it is :

$$\frac{d(\psi \psi^{*})}{dt}=\frac{i\hbar}{2m}[\frac{d\psi}{dx}\psi^{*}-\frac{d\psi^{*}}{dx}\psi]$$

By applying the RHS of the above equation :

$$\frac{i\hbar}{2m}[-A^{2}ikxe^{-i(ωt-kx)}e^{i(ωt-kx)}-A^{2}ikxe^{i(ωt-kx)}e^{-i(ωt-kx)}]$$

This gives :

$$\frac{-2iA^{2}ik\hbar}{2m}=\frac{k \hbar A^{2}}{m}$$

This is not the correct answer. :( What have I done wrong ?

In the model workings instead of A in the complex conjugate of the wave function they have written $A^{*}$. Why is this necessary since $A$ is likely to be a real number anyways ?