How to Calculate the Directional Derivative with Step-by-Step Solutions
The directional derivative is the rate at which any function changes at any specific point in a fixed direction. It is considered as a vector form of any derivative. It specifies the immediate rate of variation of the function. It particularised the vision of partial derivatives. It can be represented as :
▽uf = ▽f .( u/|u|)
= limh→0[f(k + h û) –f(k)]/h
In this article, we will discuss the concept of directional derivative in detail. We will study what is directional derivative, directional derivative definition, how to find the directional derivative, directional derivative formula, directional derivative properties etc.
\[D_{u}\] f(x,y)
Directional Derivative Definition
For a scalar function f(k) =f (k₁ , k₂,....kn), the directional derivative is defined as a function in the following manner,
▽uf = limh→0[f(k + hv) –f(k)]/h
Where v is considered as a vector along which the directional derivative f(k) is defined. Sometimes v is confined to a unit vector, or else, the definition also holds.
Vector v is derived by,
V = (v₁ , v₂,....vn)
How to Find The Directional Derivative?
The first step to find the directional derivative is to mention the direction. One method to mention the direction is with a vector u ( u₁ , u₂) that points in the direction in which we wish to find the slope. We will consider u as a unit vector. Using the directional derivative definition, we can find the directional derivative f at k in the direction of a unit vector u as
Duf (k). We can define it with a limit definition just as a standard derivative or partial derivative.
Du f (k) = limh→0[f(k +hu) –f(k)]/h
The concept of directional derivatives is quite easy to understand. Du f (k) is the slope of f(x,y) when standing at the point k and facing the direction by a unit vector (u). x and y are represented in meters then Du f (k) will be changed in height per meter as you move in the direction given by u when you are standing at the point k.
Note: Du f (k) is a matrix not a number. Directional derivative is similar as a partial derivative if u points in the positive x or positive y direction. For example if u= (1,0) then
Du f (k) = \[\partial\] f/\[\partial\]x (k). Similarly if unit vector (u) = (0,1) then,
Du f (k) = \[\partial\]f/\[\partial\]x (k)
Directional derivative properties
Some basic directional derivative properties are as follows:
The rule for a constant factor
▽v (pf) = p▽vf
Rules for the Sum
▽v (f + h) =▽vf + ▽vh
Rules for the product.
The rule for products is also known as Leibniz rule.
▽v (fh) = h▽vf + f▽vh
Chain rule
The chain rule is used when function f is differentiable at ‘a’ and g is differentiable at f(a). In such a case,
▽v ( f o h) (a) = f’(h(a)) ▽vh(a)
Directional Derivative Formula
The directional derivative formula is represented as n.▽f. Here, n is considered as a unit vector. The directional derivative is stated as the rate of change along with the path of the unit vector which is u =(p,q). The directional derivative is represented by Du F(p,q) which can be written as follows:
Du f (p,q) = limh→0[f(x + ph, y +qh) –f(p,q)]/h
Solved Examples
For the function f(m,n) = m²n., find the directional derivative of f at the point (3,2) in the direction of (2,1).
Solution: The unit vector in the direction of (2,1)
u = (2,1)/\[\sqrt{5}\]= (2/\[\sqrt{5}\], 1/\[\sqrt{5}\])
Since.,we are at the point (3,2), ( equation1) is still valid. Now we will use another value of the unit vector to get.
DU f (3, 2) = 12u1 + 9u2
= 24/\[\sqrt{5}\] + 9/\[\sqrt{5}\] = 33/\[\sqrt{5}\]
Find the directional derivative of the function f(p,q) = pqr in the direction 3i-4k. It has the point as (1,-1,1).
Solution:
Given function f(p,q) = pqr
Vector field is 3i - 4k. It has the magnitude of \[\sqrt{(3^{2}) +(-4^{2})}\] = \[\sqrt{25}\]= \[\sqrt{5}\]
The unit vector n in the direction 3i - 4k is n = 1/5(3i- 4k)
Now,we have to calculate the gradient ▽ f for calculating the directional derivative.
Hence,▽ f = qri +pri + pqk
Now, the directional derivative is
n▽ f = ⅕(3i-4k).( qri +pri + pqk)
= ⅕[ 3 × qr + 0- 4 * pq)
The directional derivative at the point (1,-1,1) is
n.▽ f = 1/5[ 3 × (-1) × (1) - 4 ×1 × (-1)
n.▽ f = 1/5
Quiz Time
Find the direction in which which the directional derivative is greater for the function
f(m,n) = 3m² 2n² - m⁴ -n⁴ at the point (1,2).
1 2(-i + j)
1 2(i - j)
1 2( i + j)
1 5( 2i + j)
-1 5(i - j)
2. The directional derivative f(m,n) = m²n³ - 2m4n at the point (1,2) in the direction 3i-4j.
1 4i + 1 2j
-96i - 56j
-152
-30.4
-32i + 14j
FAQs on Directional Derivative Explained: Concepts & Applications
1. What is a directional derivative in simple terms?
In simple terms, a directional derivative measures the rate of change of a multi-variable function at a specific point along a specific direction. Imagine you are standing on a hillside; the partial derivative would tell you the steepness if you walk directly east or north. The directional derivative tells you the steepness in any compass direction you choose to walk, for instance, northeast.
2. What is the formula to calculate the directional derivative?
The most common formula to calculate the directional derivative of a function f at a point (x, y) in the direction of a unit vector u is by using the gradient. The formula is: D_u f = ∇f ⋅ u. Here, ∇f represents the gradient vector of the function f, and '⋅' denotes the dot product between the gradient and the unit vector u.
3. What is the key difference between a partial derivative and a directional derivative?
A partial derivative measures the rate of change of a function exclusively along one of the coordinate axes (e.g., parallel to the x-axis or y-axis), assuming other variables are constant. In contrast, a directional derivative is a more general concept that measures the rate of change in any arbitrary direction, which is defined by a vector. Partial derivatives are, in fact, special cases of the directional derivative.
4. How is the gradient vector related to the directional derivative?
The gradient vector (∇f) is intrinsically linked to the directional derivative. The gradient points in the direction of the steepest ascent of the function from a given point. The directional derivative in any other direction can be thought of as the projection, or 'shadow', of the gradient vector onto that direction. This relationship is precisely captured by the formula D_u f = ∇f ⋅ u.
5. What are some real-world applications of the directional derivative?
Directional derivatives are used to model and solve problems across various fields. Key applications include:
Physics: To determine the rate of change of temperature or pressure at a point in space in a specific direction.
Engineering: To find the maximum stress or strain on a material surface along a particular axis.
Meteorology: To calculate how quickly wind speed or atmospheric pressure changes as one moves from one point to another in a specific direction.
Computer Graphics: To calculate how light and shadow change on a 3D model's surface, creating realistic visuals.
6. Why is a unit vector essential for calculating a directional derivative?
A unit vector is essential because the directional derivative is intended to measure the rate of change based on direction alone, not on the length (magnitude) of the vector defining that direction. Using a vector with a magnitude other than 1 would incorrectly scale the result. By standardising the direction vector to have a magnitude of 1, we ensure that the calculated derivative purely reflects the function's rate of change along that path.
7. In which direction is the directional derivative at its maximum value?
The directional derivative of a function at a point is maximised when the direction is the same as the direction of the gradient vector (∇f) at that point. The maximum value of the directional derivative is equal to the magnitude of the gradient vector, ||∇f||. This represents the direction of the steepest slope or greatest increase of the function.
8. What does it signify if the directional derivative at a point is zero or negative?
The sign of the directional derivative indicates the behaviour of the function in that direction:
If the directional derivative is positive, the function's value is increasing as you move in that direction.
If it is negative, the function's value is decreasing.
If it is zero, there is no instantaneous change in the function's value. This means the direction is perpendicular to the gradient and you are moving along a level curve or contour line of the surface.
