Is Gradient really the direction of steepest ascent? I want to intuitively understand why the gradient gives you the direction of the **steepest** ascent of a function. Apart from the already posted questions, my confusion arises from the fact that we form the gradient vector from the derivative of each dimension **separately**. Then take the vector consisting of both (for 2D) derivatives take it as the steepest ascent. What if in both directions the derivative is say $5$, so our vector will be $45$ degrees from both axis, But in that direction specifically the function goes down ? If it's not clear what I'm confused with, consider this function represented as an image : $$ \begin{pmatrix}100&5&-100\\\0&\textit{0}&5\\\0& 0& 0\end{pmatrix}$$ at _0_ , it makes sense that the derivative is $5$ in $x$ and in $y$, but a vector of $(5,5)$ goes to a direction that's not a steepest ascent. Does this have to do with the differentiability of the function ? what am I missing ?

What will help your intuition the most is remembering that the derivative (the gradient) is a local feature, it only depends on what the function is _at that point_ , and not any distance away.

You may be visualizing a function which buckles down in the gradient direction, so it's not the steepest ascent some distance away -- but at the point where you find the tangent plane it is the steepest ascent for at least a very small distance.

At a point where a function is differentiable, the function is almost planar in a very, very small region around that point. Remember to visualize the local region as nearly a plane, and your intuition will be happier with the gradient.