Why is the gradient of $f(x)= x$ the vector $(1,0,0)$?

Question

Why is the gradient of $f(x)= x$ the vector $(1,0,0)$? This would mean that at every point on the straight line $y = x$, I should go horizontally to experience the maximum change in slope. I was expecting the gradient at $(x,y)$ to be the the position vector of $(x,y)$ as it makes more sense. How does it make sense that the maximum change in slope will be attained by trying to move horizontally?

Answer 1

The gradient of $f(x)=x$ is not $(1,0,0)$. It's just $1$, which means if you want to move in the direction of greatest increase of $f$, you should move in the $+1$ direction on the $x$-axis.

The gradient of $f(x,y,z)=x$ is $(1,0,0)$. This function has domain $\mathbb{R}^3$, so its graph is in $4$-dimensional space (making it difficult to picture). You can imagine it as placing a "label" at every point $(x,y,z) \in \mathbb{R}^3$ where the label says the point's $x$-value. If we want to move in the direction of maximum increase of the labels, we should move in the $(1,0,0)$ direction (i.e., in the positive $x$-direction).