This question arises from a physics question, where in the working the author wrote $$\int^b_a \frac{\partial}{\partial t}\left[K(t-\frac{x}{c}-u)\right]\, du=-\int^b_a \frac{\partial}{\partial u}\left[K(t-\frac{x}{c}-u)\right]\, du,$$ where $t$, $x$ and $u$ are variables and $c$ is a constant. $K(t')$ is just a function. The integral on the right-hand side can then be evaluated using the FTC.
Why is this change in derivative variable allowed? At first I thought it was the chain rule, but then the right-hand side is missing a factor of $\frac{\partial u}{\partial t}$. From what I can see, there is no explicit relation between $u$ and $t$. Can someone explain this please?
$$\frac{\partial}{\partial t}\left[K(t-\frac{x}{c}-u)\right]=K'(t-\frac{x}{c}-u)$$
$$\frac{\partial}{\partial u}\left[K(t-\frac{x}{c}-u)\right]=K'(t-\frac{x}{c}-u)(-1)=-K'(t-\frac{x}{c}-u)$$