Calculus
Textbooks
Boundless Calculus
Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
Partial Derivatives
Calculus Textbooks Boundless Calculus Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus Partial Derivatives
Calculus Textbooks Boundless Calculus Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
Calculus Textbooks Boundless Calculus
Calculus Textbooks
Calculus
Concept Version 12
Created by Boundless

Directional Derivatives and the Gradient Vector

The directional derivative represents the instantaneous rate of change of the function, moving through $\mathbf{x}$ with a velocity specified by $\mathbf{v}$.

Learning Objective

  • Describe properties of a function represented by the directional derivative


Key Points

    • The directional derivative is defined by the limit $\nabla_{\mathbf{v}}{f}(\mathbf{x}) = \lim_{h \rightarrow 0}{\frac{f(\mathbf{x} + h\mathbf{v}) - f(\mathbf{x})}{h}}$.
    • If the function $f$ is differentiable at $\mathbf{x}$, then the directional derivative exists along any vector $\mathbf{v}$, and one gets $\nabla_{\mathbf{v}}{f}(\mathbf{x}) = \nabla f(\mathbf{x}) \cdot \mathbf{v}$ .
    • Many of the familiar properties of the ordinary derivative hold for the directional derivative.

Terms

  • gradient

    of a function $y=f(x)$ or the graph of such a function, the rate of change of $y$ with respect to $x$; that is, the amount by which $y$ changes for a certain (often unit) change in $x$.

  • chain rule

    a formula for computing the derivative of the composition of two or more functions.


Full Text

The directional derivative of a multivariate differentiable function along a given vector $\mathbf{v}$ at a given point $\mathbf{x}$ intuitively represents the instantaneous rate of change of the function, moving through $\mathbf{x}$ with a velocity specified by $\mathbf{v}$. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the coordinate curves, all other coordinates being constant.

Definition

The directional derivative of a scalar function $f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n)$ along a vector $\mathbf{v} = (v_1, \ldots, v_n)$ is the function defined by the limit:

$\displaystyle{\nabla_{\mathbf{v}}{f}(\mathbf{x}) = \lim_{h \rightarrow 0}{\frac{f(\mathbf{x} + h\mathbf{v}) - f(\mathbf{x})}{h}}}$

If the function $f$ is differentiable at $\mathbf x$, then the directional derivative exists along any vector $\mathbf v$, and one has $\nabla_{\mathbf{v}}{f}(\mathbf{x}) = \nabla f(\mathbf{x}) \cdot \mathbf{v}$, where the $\nabla f(\mathbf{x})$ is the gradient vector and $\cdot$ is the dot product. At any point $\mathbf x$, the directional derivative of $f$ intuitively represents the rate of change of $f$ with respect to time when it is moving at a speed and direction given by $\mathbf v$ at the point $\mathbf x$. The name "directional derivative" is a bit misleading since it depends on both the length and direction of $\mathbf v$.

We can imagine the directional derivative $\nabla_{\mathbf{v}}{f}(\mathbf{x})$ as the slope of the tangent line to the 2-dimensional slice of the graph of $f$ that lies parallel to the vector $\mathbf{v}$. However, this slice will be stretched or compressed horizontally unless $\mathbf{v}=1$.

Gradient of a Function

The gradient of the function $f(x,y) = −\left((\cos x)^2 + (\cos y)^2\right)$ depicted as a projected vector field on the bottom plane. Directional derivative represents the rate of change of the function along any direction specified by $\mathbf{v}$.

Properties

Many of the familiar properties of the ordinary derivative hold for the directional derivative.

The Sum Rule

$\nabla_\mathbf{v} (f + g) = \nabla_\mathbf{v} f + \nabla_\mathbf{v} g$

The Constant Factor Rule

For any constant $c$, $\nabla_\mathbf{v} (cf) = c\nabla_\mathbf{v} f$.

The Product Rule (or Leibniz Rule)

$\nabla_\mathbf{v} (fg) = g\nabla_\mathbf{v} f + f\nabla_\mathbf{v} g$

The Chain Rule

If $g$ is differentiable at $p$ and $h$ is differentiable at $g(p)$, then $\nabla_\mathbf{v} h\circ g (p) = h'(g(p)) \nabla_\mathbf{v} g (p)$.

[ edit ]
Edit this content
Prev Concept
The Chain Rule
Maximum and Minimum Values
Next Concept
Subjects
  • Accounting
  • Algebra
  • Art History
  • Biology
  • Business
  • Calculus
  • Chemistry
  • Communications
  • Economics
  • Finance
  • Management
  • Marketing
  • Microbiology
  • Physics
  • Physiology
  • Political Science
  • Psychology
  • Sociology
  • Statistics
  • U.S. History
  • World History
  • Writing

Except where noted, content and user contributions on this site are licensed under CC BY-SA 4.0 with attribution required.