TITLE: Why is gradient in the direction of ascent but not descent?
QUESTION [14 upvotes]: I understand that differentiation of a function ($\mathbb{R} \rightarrow \mathbb{R} $) at a point is the rate of change in the output for a slight nudge in the input. And this rate of change could be negative or positive. There is no concept of direction for the single-variable function as obvious.
Now, my doubt is in the case of the multivariate function ($\mathbb{R}^n \rightarrow \mathbb{R}$) where differentiation is a gradient. And this gradient representing partial differentiation w.r.t. to each basis becomes a direction. This direction is a direction of ascent but not descent, why?. Why it is a direction is of ascent. My question is not at all related to steepest ascent, about which one can find many answers on this forum and read elaborately at this link. An intuitive explanation would be preferable than mathematical at this link.

REPLY [5 votes]: "There is no concept of direction for the single-variable function as obvious."
False.
When the univariate derivative is positive, the function increases in the direction of increasing input; when negative, in the direction of decreasing input.  There are only two possible directions, but whichever it is, the derivative does point in the direction in which the function increases.