Partial and Directional Derivatives

This page is a sub-page of our page on Calculus of Several Real Variables.

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Directional derivative:

Let $f:\mathbf{R}^n \rightarrow \mathbf{R}$ and choose $\, e \in \mathbf{R}^n$ with $||e|| = 1.$ If the limit

$\lim\limits_{\epsilon \rightarrow 0} \dfrac{f(a + \epsilon e) - f(a)}{\epsilon}$

exists, this limit is called the directional derivative of the function f at the point $a \in \mathbf{R}^n$ in the direction $e \in \mathbf{R}^n$, and it is denoted by $D_e f(a).$

OBS: $D_{e_i} f(a) = \frac{\partial }{\partial x_i} f(a).$

Interactive simulation of Directional Derivative.

Directional derivative:

Partial derivatives:
The directional derivatives along one of the coordinate axes is called a partial derivative. They can be thought of as measuring the rate of change of a function along the direction of a specific one of its variables, while all the other variables are kept constant:

Definition: