From time to time, I suddenly get confused with a change of variables in a partial derivative.

Here, I am trying to perform a change of variables $(x,t) \mapsto (\xi, \eta)$ where

$$\xi = t \qquad \qquad \text{and} \qquad \qquad \eta = x+t$$

The question is, how to compute $$\frac{\partial u}{\partial t}$$ in the new coordinate system?

Intuitively, since $\xi = t$ (or rather $t=\xi$), we should have

$$\frac{\partial u}{\partial t} = \frac{\partial u}{\partial \xi}$$

However, applying the chain rule for partial derivatives, we instead get

$$\frac{\partial u}{\partial t} = \frac{\partial u}{\partial \xi}\frac{\partial \xi}{\partial t} + \frac{\partial u}{\partial \eta}\frac{\partial \eta}{\partial t} = \frac{\partial u}{\partial \xi}(1) + \frac{\partial u}{\partial \eta}(1) = \frac{\partial u}{\partial \xi} + \frac{\partial u}{\partial \eta}$$

So which one is correct?