Directional Derivative Formula Definition and Solved Examples
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 in Multivariable Calculus
1. What is a directional derivative?
The directional derivative measures the rate at which a function changes at a point in a specified direction. For a function f(x, y), it tells you how fast the function increases or decreases when moving in a given direction vector. Mathematically, it represents the slope of the function along that direction and generalizes the idea of a partial derivative.
2. What is the formula for the directional derivative?
The formula for the directional derivative of f in the direction of a unit vector u is Duf = ∇f · u. Here:
- ∇f is the gradient vector of f
- u is a unit direction vector
- “·” denotes the dot product
3. How do you calculate a directional derivative step by step?
To calculate a directional derivative, compute the gradient and take its dot product with a unit direction vector.
- 1. Find the gradient: ∇f = (fx, fy)
- 2. Convert the direction vector into a unit vector
- 3. Compute Duf = ∇f · u
4. Why must the direction vector be a unit vector?
The direction vector must be a unit vector because the directional derivative measures rate of change per unit length. If the vector is not normalized, the result will be scaled incorrectly. To convert a vector v into a unit vector, divide it by its magnitude: u = v / |v|.
5. What is the relationship between the gradient and the directional derivative?
The gradient vector determines the directional derivative through the dot product formula Duf = ∇f · u. The gradient points in the direction of maximum increase of the function, and its magnitude gives the maximum possible directional derivative at that point.
6. What is the maximum value of a directional derivative?
The maximum value of the directional derivative at a point is the magnitude of the gradient, written as |∇f|. This occurs when the direction vector is the same as the gradient direction. The minimum value is −|∇f|, occurring in the opposite direction.
7. What is the difference between a partial derivative and a directional derivative?
A partial derivative measures change in one coordinate direction, while a directional derivative measures change in any chosen direction. Partial derivatives are special cases of directional derivatives taken along standard unit vectors like (1,0) or (0,1).
8. Can you give an example of a directional derivative?
Yes, for f(x, y) = xy, the directional derivative at (1,2) in direction (3,4) can be computed using the gradient.
- ∇f = (y, x)
- At (1,2): ∇f = (2,1)
- Unit vector in direction (3,4) is (3/5, 4/5)
- Duf = (2,1) · (3/5, 4/5) = 10/5 = 2
9. When does the directional derivative equal zero?
The directional derivative equals zero when the direction vector is perpendicular to the gradient vector. Since Duf = ∇f · u, the dot product becomes zero when the angle between them is 90°, meaning there is no change in that direction.
10. What are the applications of directional derivatives?
The directional derivative is used to measure rates of change in multivariable calculus and applied mathematics. Common applications include:
- Finding maximum increase using the gradient
- Optimization problems in economics and engineering
- Surface slope analysis in physics
- Machine learning gradient-based methods
