Consider the scalar-valued function $y = 2\mathbf{x}^{ op}\mathbf{x}$. When differentiated with respect to the column vector $\mathbf{x}$, the resulting gradient is $4\mathbf{x}$. In deep learning frameworks, after allocating gradient memory and computing $y$, calling the backward pass function calculates this gradient and stores it in the vector's gradient attribute.