I've come across using partial derivative notation for taking the partial derivative of a function f with respect to a vector x. I've never seen this before. It is also being referred to as a gradient. However, I have only seen gradients where all variables in the space are featured in the result vector. In this case, the result is a vector but not with components representing each dimension in the space. On wikipedia I've seen this referred to as matrix calculus notation. I would like to know a bit more about this in broad terms. For instance, for a space x1, x2, x3, x4 if I take the partial derivative with respect to a vector x1,x2 is that result vector valued function pointing in the direction of steepest ascent similar to a gradient but only for x1 and x2? Any other pointers appreciated.