---
tag: 0ABK9V9Y64
kind: concept
name: Gradient
aliases: [gradient-vector, grad]
msc: [26B12]
uses: [09EGC6F8PT, 0HHHDCXZV4]
sources:
  - url: https://en.wikipedia.org/wiki/Gradient
    locator: lead section
same_as:
  wikidata: Q173582
  wikipedia: Gradient
---

## Summary

Let $f$ be a [[differentiable|0HHHDCXZV4]] real-valued function on an open subset of $\mathbb{R}^n$. Its gradient at a point $a$ is the vector $\nabla f(a)$ of its [[partial derivatives|09EGC6F8PT]] at $a$.
It points in the direction in which $f$ increases fastest, and its length is that fastest rate of increase.
The derivative of $f$ at $a$ is the dot product with the gradient: $Df(a)(v) = \nabla f(a) \cdot v$.