Determinant of a coherent sheaf over a smooth projective variety

We know a coherent sheaf $E$ over a smooth projective variety $X$ admits a finite locally free resolution. $0\longrightarrow E_n\longrightarrow E_{n-1}\longrightarrow\cdots\longrightarrow E_0\longrightarrow E\longrightarrow 0$.

So we define the determinant of $E$ to be $\textrm{det}(E)=\otimes\textrm{det}({E}_i)^{(-1)^i}$, this is a line bundle.

My doubt is as follows : How do we know that this is independent of the choice of locally free resolution? Any clarification or reference would really help.