What is the derivative of the sum of a vector with respect to the vector?

Question

If X is a vector, what is the derivative of : $\frac{d}{dX} 1^TX$ ?

where $1^T$ is the summation operation e.g. $1^TX = \sum_n x_n $

Answer 1

Just work it out with multivariable calculus by noting that the derivative of a linear term is constant.

$\frac{d}{dX}1^TX = 1^T\frac{d}{dX}X = 1^T I = 1$

Where $1$ is a vector of $1s$

Answer 2

We call $\frac{d}{dX}=\nabla$ the gradient operator in multivariable calculus. The gradient here is given by a row vector with entries $$\left[\frac{\partial (1^T X)}{\partial x_i}\right]_i = \left[ \frac{\partial (\sum_n x_n)}{\partial x_i} \right]_i = \left[1\;1 \cdots\right] = \frac{d (1^T X)}{dX}\,\text{.}$$