quantum mechanics - Probability current density : Isn't there something wrong with this proof?

Is there something wrong with the following proof (see below)? To me, it seems like the third line should show

$$\frac{dP_{ab}}{dt}=-\int_a^b \frac{\partial}{\partial t}J(x,t)dx$$ Am I missing something obvious?

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Answer

If we look at line 2, we have an integral set equal to:

We then must note that $J(x,t)$ is defined as the negative of:

So the last equation on line 2 should be: $-\frac{\partial J(x,t)}{\partial x}$ as opposed to $-\frac{\partial J(x,t)}{\partial t}$.