Directional Derivative

Directional Derivative measures how a function changes along a specified direction at a given point, providing insights into its rate of change in that direction. Directional Derivative can be defined as:

D_v(f) = ∇f · v

In this article, we will learn about the directional derivative, its meaning, definition, steps to calculate the directional derivative, directional derivative in three dimensions, and others in detail.

What is a Directional Derivative?

Directional derivative provides valuable information about how a function changes concerning specific directions in space, making it useful in various fields such as physics, engineering, and economics.

Directional Derivative Definition

A directional derivative is a derivative of a function in the direction of a specified vector. It measures how the function changes as you move along a particular direction in its domain.

Directional Derivative Formula

Calculating the directional derivative involves the dot product of the gradient and the direction vector. In simple terms, for a function f, the directional derivative D_v(f) is given by:

D_v(f) = ∇f · v

where:

• ∇f represents Gradient of Function
• v is Direction Vector Along which we Want to Find Derivative

How to Calculate Directional Derivative

To calculate the directional derivative of a function at a given point in a specific direction, follow these steps:

Step 1: Find the Gradient

Compute the gradient (∇f) of the function. The gradient is a vector that points in the direction of the steepest increase of the function.

Step 2: Normalize Direction Vector

Normalize the direction vector (v) to ensure it has a length of 1. This is done by dividing each component of the vector by its magnitude.

Step 3: Dot Product

Take the dot product of the normalized direction vector and the gradient. The dot product is obtained by multiplying corresponding components of the two vectors and then summing them up.

D_v​(f) =∇f⋅v

Step 4: Evaluate at a Point: Plug in the coordinates of the point where you want to find the directional derivative into the gradient and the normalized direction vector.

D_v​(f)(a, b) = ∇f(a, b)⋅v

Directional Derivative Formula in Vector Calculus

In three dimensions or vector calculus, the directional derivative measures how a function changes along a specific direction in space. Mathematically, it is denoted as ∇v​f or D_v​(f), where f is the function and v is the direction vector.

The directional derivative in three dimensions is computed using the dot product between the gradient of the function and the normalized direction vector:

∇_v​f = ∇f⋅v

where:

• ∇f represents Gradient of Function
• v is Unit Vector Indicating Direction in which Derivative is to be Calculated

Directional Derivative in Different Coordinate Systems

Directional derivatives helps in analysing how functions change along specific directions at given points, and they are applicable in various coordinate systems such as Cartesian, cylindrical, and spherical coordinates.

Directional Derivative in Cartesian Coordinates

In Cartesian coordinates, directional derivatives are calculated using the gradient operator (∇), which is a vector representing the rate of change of a scalar function in each coordinate direction. For a function f(x, y, z), the directional derivative in the direction of a unit vector u = (u₁, u₂, u₃) at a point P(x₀, y₀, z₀) is given by the dot product of the gradient of (f) and (u):

D_uf = ∇ f · u

Directional Derivative in Cylindrical Coordinates

In cylindrical coordinates (ρ, ϕ, z), where (ρ) represents the radial distance, (ϕ) the azimuthal angle, and (z) the vertical position, directional derivatives are computed by transforming the gradient operator (∇) to cylindrical coordinates. The directional derivative in the direction of a unit vector u = (u_ρ​, u_ϕ​, u_z​) at a point P(ρ_0​, ϕ_0​, z₀) is given by:

D_uf = ∇ f · u

Directional Derivative in Spherical Coordinates

In spherical coordinates (r, θ, ϕ), (r) represents the radial distance, (θ) the polar angle, and (ϕ) the azimuthal angle, directional derivatives are similarly computed using the gradient operator (∇) transformed to spherical coordinates. The directional derivative in the direction of a unit vector u = (u_r​, u_θ​, u_ϕ) at a point P(r₀​, θ₀​, ϕ₀) is given by:

D_uf = ∇ f · u

Properties of Directional Derivative

Directional derivatives possess certain properties that can be described as follows:

Rule for Constant Factor: If we have a constant (k), then the directional derivative of (kf) in the direction of (v) is given by (k) times the directional derivative of (f) in the same direction.

∇_v(kf) = k ∇_vf

Rule for Sum: The directional derivative of the sum of two functions (f) and (g) in the direction of (v) is the sum of their individual directional derivatives in the same direction.

∇_v(f + g) = ∇_vf + ∇_vg

Rule for Product (Leibniz’s Rule): For the product of two functions (f) and (g), the directional derivative in the direction of (v) is given by the product of the directional derivative of one function and the other function, plus the product of the original functions with their respective directional derivatives.

Chain Rule: When applying the chain rule, if (f) is differentiable at point (a) and (g) is differentiable at f(a), then the directional derivative of their composition is the product of the derivative of the outer function evaluated at the inner function’s value and the directional derivative of the inner function.

Linearity and Directional Derivative

Linearity in the context of directional derivatives refers to the property where the directional derivative of a linear combination of functions is equal to the same linear combination of their directional derivatives. Mathematically, this can be expressed as follows:

Let f and g be differentiable functions defined in a neighborhood of a point a, and let k₁​ and k₂​ be constants. Then, the directional derivative of k_1​f+k_2​g in the direction of v can be written as:

∇_v​(k₁​f + k₂​g) = k₁​∇_v​f + k₂​∇_v​g

This property essentially states that the directional derivative behaves linearly with respect to linear combinations of functions. It’s a fundamental property that allows for easier computation and understanding of how directional derivatives behave under combinations of functions.

Directional Derivative Gradient

Gradient for the function f(x,y) is defined as:

▽f = ▽f(x, y) = ∂f/∂xi + ∂f/∂yj

Directional derivative of the function is calculated by taking the dot product of the gradient and the given unit vector. For a function f(x, y, z) and unit vector u(u₁, u₂, u₃) then is directional derivative gradient is calculated as:

D_uf = ▽f.u = ∂f/∂x u₁ + ∂f/∂y u₂ + ∂f/∂z u₃

Difference Between Directional Derivative and Partial Derivative

Difference between directional derivative and partial derivative can be understood from the table below:

Directional Derivative vs Partial Derivative
Property	Directional Derivative	Partial Derivative
Definition	Measures the rate of change of a function in a specific direction from a given point.	Measures the rate of change of a function with respect to a single variable, holding others constant.
Symbol	∇v​f	∂f/∂x, ∂f/∂y, etc
Direction	Requires a direction vector v	Taken along one coordinate axis (e.g., x or y)
Notation	∇v​f or Dv​(f)	∂f/∂x or ∂f/∂y
Geometric Interpretation	Represents the slope of the function in a specified direction.	Represents the slope of the function along one coordinate axis.
Example	Rate of change of temperature in the direction of the wind.	Rate of change of temperature with respect to time.
Calculation	Involves the dot product of the gradient and the direction vector.	Involves finding the partial derivatives with respect to each variable.
Multiple Variables	Applicable to functions of multiple variables.	Applicable to functions of multiple variables.

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Directional Derivative Examples

Example 1: Compute the directional derivative of the function f(x, y) = x² + 3y at the point P(1, 2) in the direction of the vector v=⟨1, −1⟩.

Solution:

To compute the directional derivative of the function f(x, y) = x² + 3y at the point P(1, 2) in the direction of the vector v = ⟨1, −1⟩, we use the following formula:

D_v​f = ∇f⋅v

First, let’s find the gradient of f:

So, ∇f=(2x,3).

Now, evaluate ∇f at the point P(1,2):

∇f(1,2)=(2(1),3)=(2,3)

Next, we compute the dot product of ∇f(1,2) and v:

∇f(1,2)⋅v = (2, 3)⋅⟨1, −1⟩ = 2⋅1 + 3⋅(−1) = 2 − 3 = −1

Therefore, the directional derivative of f(x, y) = x² + 3y at the point P(1, 2) in the direction of the vector v = ⟨1, −1⟩ is D_v​f = −1.

Example 2: Find the directional derivative of g(x,y,z) = xyz at the point Q (2, −1, 3) in the direction of v = ⟨−2, 1, 2⟩.

Solution:

To find the directional derivative of the function g(x ,y, z) = xyz at the point Q(2, −1, 3) in the direction of the vector v=⟨−2, 1, 2⟩, we use the formula:

D_v​g = ∇g⋅v

First, let’s find the gradient of g:

So, ∇g = (yz, xz, xy).

Now, evaluate ∇g at the point Q(2, −1, 3):

∇g(2, −1, 3) = (yz, xz, xy) = (−3, 6, −2)

Next, we compute the dot product of ∇g(2, −1, 3) and v:

∇g(2,−1,3)⋅v = (−3,6,−2)⋅⟨−2,1,2⟩

∇g(2,−1,3)⋅v = (−3)(−2)+(6)(1)+(−2)(2)

∇g(2,−1,3)⋅v = (−3)(−2)+(6)(1)+(−2)(2)

∇g(2,−1,3)⋅v = 6+6−4

∇g(2,−1,3)⋅v = 6+6−4

∇g(2,−1,3)⋅v = 8

Therefore, the directional derivative of g(x, y, z) = xyz at the point Q(2, −1, 3) in the direction of the vector v=⟨−2, 1, 2⟩ is D_v​g = 8.

Example 3: Calculate the directional derivative of h(x, y) = e^xy at the point R(0, 1) in the direction of v = ⟨1, 1⟩.

Solution:

To calculate the directional derivative of the function h(x, y) = e^xy at the point R(0, 1) in the direction of the vector v=⟨1, 1⟩, we use the formula:

D_v​h = ∇h⋅v

First, let’s find the gradient of h:

∇h = (ye^xy, xe^xy)

Now, evaluate ∇ℎ∇h at the point R(0, 1):

∇h(0, 1) = (ye^xy, xe^xy) = (1⋅e⁰, 0⋅e⁰) = (1, 0)

Next, we compute the dot product of ∇h(0, 1) and v:

∇h(0,1)⋅v = (1,0)⋅⟨1,1⟩

= (1)(1) + (0)(1)

∇h(0,1)⋅v = (1)(1) + (0)(1) = 1

Therefore, the directional derivative of h(x, y) = e^xy at the point R(0, 1) in the direction of the vector v=⟨1, 1⟩ is D_v​h = 1

Practice Problems on Directional Derivative

Problem 1: Determine the directional derivative of f(x, y, z) = x² + 2y − 3z at the point P(1, 2, −1) in the direction of v=⟨−1, 2, 3⟩.

Problem 2: Compute the directional derivative of g(x, y) = cos(xy) at the point Q(π, 2) in the direction of v=⟨−1, 1⟩.

Problem 3: Find the directional derivative of h(x, y, z) = ln(x² + y² + z²) at the point R(1, −1, 2) in the direction of v=⟨1, 2, −1⟩.

Problem 4: Calculate the directional derivative of f(x, y) = x³ + y² at the point P(−2, 3) in the direction of v=⟨2, 3⟩.

Problem 5: Determine the directional derivative of g(x, y, z) = e^xyz at the point Q(1, 1, 1) in the direction of v=⟨1, −1, 2⟩.

Summary – Directional Derivative

The directional derivative is a crucial concept in multivariable calculus that measures how a function changes along a specific direction at a given point in its domain. It is defined as the dot product of the gradient of the function and a direction vector, indicating the rate of change of the function in that direction. To calculate it, one must first compute the gradient of the function, normalize the direction vector to unit length, and then perform the dot product of these two vectors. This concept extends to various coordinate systems, including Cartesian, cylindrical, and spherical coordinates, allowing for its application in a wide range of fields such as physics, engineering, and economics. The directional derivative has several properties, including linearity, which facilitates the analysis of linear combinations of functions, and it plays a significant role in understanding the behavior of functions in multidimensional spaces.

FAQs on Directional Derivative

What is the directional derivative?

Directional derivative is a measure of how a function changes in a particular direction at a given point. It represents the rate of change of the function along the specified direction.

Is the gradient the directional derivative?

No, the gradient is not the directional derivative. The gradient is a vector that points in the direction of the steepest ascent of a scalar field, while the directional derivative is a scalar representing the rate of change of the function in a specified direction.

What is differentiable and directional derivative?

A function is differentiable if it has well-defined derivatives at every point in its domain. The directional derivative, on the other hand, measures the rate of change of the function along a specified direction at a given point.

What is the difference between divergence and directional derivative?

Divergence and directional derivative are different concepts in vector calculus. Divergence represents the measure of the spreading of a vector field at a point, while the directional derivative measures how a scalar field changes along a particular direction at a given point.

What is the first principle of directional derivative?

The first principle of directional derivative states that the directional derivative of a function in the direction of a vector v at a point p is equal to the dot product of the gradient of the function at p and the unit vector in the direction of v.

What is the maximum directional derivative formula?

The maximum directional derivative formula gives the maximum rate of change of a function in a given direction. It is obtained by taking the dot product of the gradient of the function and the unit vector in the specified direction.

Is directional derivative a vector or scalar?

The directional derivative is a scalar quantity, representing the rate of change of a function in a particular direction. While it is computed using vector operations, the directional derivative itself is not a vector.

What is directional derivative vs gradient?

Directional derivative is the rate of change of a function in any given direction. Whereas the gradient is the formula used to calculate the directional derivative.