In my QFT course, we are doing some infinitesimal transformations of scalar fields.
We do the following :
$$ \phi'(x')=\phi'(x+\delta x) =\phi'(x)+\delta x^\mu \partial_\mu \phi(x)$$
But i don't get why it is $\partial_\mu \phi(x)$ and not $\partial_\mu \phi'(x)$ ?
Why would the derivative of $\phi'$ be the same as the derivative of $\phi$ ?
Is it because $\phi'=\phi+\delta \phi$ and we only keep the first order terms ? So $\delta x^\mu \partial_\mu \phi'(x)=\delta x^\mu \partial_\mu (\phi+\delta \phi)(x)=\delta x^\mu \partial_\mu \phi(x)$ at first order ?