When differentiating a vector-valued function, such as the element-wise product $y = x * x$, deep learning frameworks require the output to be reduced to a scalar to compute a gradient vector of the same shape as the input. For example, given the input vector $x = [0, 1, 2, 3]$, the element-wise product yields $y = [0, 1, 4, 9]$. Reducing this output by summing its elements gives $\sum_i x_i^2$, the gradient of which is $2x$, resulting in the vector $[0, 2, 4, 6]$. Frameworks handle this reduction differently: TensorFlow and MXNet implicitly sum the output vector, PyTorch requires passing a reduction vector of ones via the gradient argument (e.g., y.backward(gradient=torch.ones(len(y)))) or using an explicit sum, and JAX requires the function to explicitly return a scalar sum before applying the grad transform.