What is Directional Derivative in Vector Calculus?
Introduction
Directional derivative is a rate of change of a function at a point in its direction. It follows the concept of partial derivatives. In this article, you will learn more aboutdirectional derivative. You will also learn why directional derivatives are necessary.
Partial Derivative Definition
The partial derivative of a function defines the derivative with respect to more than one variable. In this differentiation method, when we are taking the derivative of a function with two variables, we find the derivative with respect to one variable while keeping the other one as constant.
For a function f(x,y), the partial derivative with respect to x and y is defined as
$$ Df\;=\;f_x+f_y $$
And,
$$f_x\;=\;\frac{df}{dx}\;\;,\;\; f_y\;=\;\frac{df}{dy}$$
Directional Derivative Definition
The rate of change of a function at a particular point, let suppose P, in the direction of some vector V. Let f(x,y) be a function at a point (xo , yo), in the direction of a vector v= (a,b) then the direction derivative of this function is defined by the partial derivative of f(x,y) at point (xo , yo) such that
$$ D_vf\;(x_o,y_o)\;=\;f_x\;(x_o,y_o)a\;+\;f_y\;(x_o,y_o)b $$
The directional derivative is denoted by Dvf (xo,yo).
The concept of directional derivative is based on the partial derivative of a function. Since the partial derivative finds the derivative of a function with respect to one variable and takes the other as constant. While the directional derivative finds the derivative of a function when x and y both are changing. There is a difference between derivative and directional derivative.
Properties of Directional Derivative
Just like derivatives, there are some properties or rules for directional derivatives also. Suppose f(x,y) and g(x,y) are two functions, so their directional derivatives satisfy the following rules
Rule for Constant Factor
Let k be a constant, then;
$$▽_v(kf)\;= \;k▽_vf$$
Rule for Sum
The sum is distributive.
$$▽_v\;(f\;+\;g)\;= \;▽_vf\;+\;▽_vg$$
Rule for Product
$$▽_v(fg)\;=\;g\;▽_vf\;+\;f\;▽_vg$$
This is also known as Leibniz’s rule.
Chain Rule
It applies when f is differentiable at ‘a’ and g is differentiable at f(a). In such a case,
$$▽_v(f\;o\;g)(a)\;=\;f'(g(a))\;▽_vg(a)$$
Why do we need directional derivatives?
In calculus and vector analysis, derivatives are important to find instantaneous rate of change. But the derivative does not tell how the change will occur if there is a vector in the direction of instantaneous change of a function.
So, here we need a way to consistently find the rate of change of a function along a vector. The directional derivative helps to find the rate of change in the given direction.
Although the maximum part of calculus is based on derivatives, it is still unable to tell us how to calculate the rate of change when both variables vary with the change. i.e. the rate of change of a function f(x,y) is due to increase or decrease in x and y both. Directional derivative is an advanced term of derivatives that handles this kind of change easily.
How to find directional derivatives of a function?
To understand how to calculate directional derivatives of a function at a given point, we should have prior knowledge of partial derivatives. It is because the formula of directional derivative is the product between the given vector v and the partial derivative at the given point P. The formula is
$$ D_vf\;(x_o,y_o)\;=\;f_x(x_o,y_o)a\;+\;f_y(x_o,y_o)b$$
Where,
$$v\;=\;(a,b)$$
$$ P\;=\;(x_o,y_o)$$
Here is an example showing how to calculate a directional derivative. You can also find the directional derivative with an agle.
Directional Derivative Example
Consider the function of the graph shown in the previous section: f(x,y) = x2 + y2 and the point on the graph P = (1,1). What is the directional derivative of the function f(x,y) at the point P in the direction of v = <1,1>?
Given that,
$$f(x,y)\;=\;x^2\;+\;y^2$$
To calculate directional derivative, we need to first calculate partial derivative. So,
$$f_x\;(x,y)\;=\;2x \; ,\; f_y\;(x,y)\;=\;2y$$
At the point P = (1,1)
$$f_x(1,1)\;=\;21\;=\;2\; ,\; f_y(1,1)\;=\;21\;=\;2$$
And
$$|v|\;=\;\sqrt{1^2\;+\;1^2}\;=\;\sqrt{2}$$
So the unit vector is
$$ v\;=\;\frac{v}{|v|}\;=<\frac{1}{\sqrt 2}\;,\;\frac{1}{\sqrt 2}\;> $$
Hence the directional derivative in the direction of v at point P is
$$ D_vf\;(1,1)\;=\;f_x(1,1)\;\frac{1}{\sqrt 2}\;+\;f_y\;(1,1)\;\frac{1}{\sqrt 2} $$
$$ D_vf\;(1,1)\;=\;\frac{2}{\sqrt2}\;+\;\frac{2}{\sqrt 2}\;=\; \frac{4}{\sqrt 2} $$
FAQ’s
Is directional derivative a scalar or vector?
The directional derivative is a product of a unit vector and the partial derivative of a function at a point P. So, it is a scalar because dot product always results in a scalar quantity. For this property, the dot product is also known as scalar product.
What is directional derivative?
Let f(x,y) be a function at a point (xo,yo), in the direction of a vector v= then the direction derivative of this function is defined by the partial derivative of f(x,y) at point (xo,yo) such that
$$ D_vf\;(x_o,y_o)\;=\;f_x(x_o,y_o)a\;+\;f_y(x_o,y_o)b$$
Where a and b are the components of the unit vector of v. fx(xo,yo) and fy(xo,yo) are partial derivatives at (xo,yo).
Why is directional derivative a dot product?
For a function f(x,y), the partial derivative with respect to x gives the rate of change of f in the x direction and the partial derivative with respect to y gives the rate of change of f in the y direction. By the formula of directional derivative,
$$ D_vf\;(x_o,y_o)\;=\;f_x(x_o,y_o)a\;+\;f_y(x_o,y_o)b$$
It is clear that the directional derivative is the dot product of the partial derivatives and the vector v.
What is the formula of directional derivative?
The formula of directional derivative contains the dot product between the gradient vector and the unit vector at a given point, that is,
$$ D_vf\;(x_o,y_o)\;=\;f_x(x_o,y_o)a\;+\;f_y(x_o,y_o)b$$