In Sean Carroll's GR book, pg 84, the exterior derivative $d$ is defined as $$(dA)_{\mu_1\mu_2...\mu_{p+1}} = (p+1) \partial_{[\mu_1}A_{\mu_2...\mu_{p+1}]}\tag{2.76}$$ where $A$ is a $p$-form and the RHS is the appropiately normalized and antisymmetrized partial derivative.
I know that to antisymmetrize a tensor, we mean something like $$t_{[ab]}=\frac{1}{2}(t_{ab}-t_{ba}).$$
But the partial derivative of a tensor is generally not a tensor. So what does it mean to antisymmetrize the partial derivative of a tensor? In other words, what does $\partial_{[\mu_1}A_{\mu_2...\mu_{p+1}]}$ mean?