How to find Directional Derivative of a Function?

Introduction

Directional derivative is one of the key concepts of calculus in which we calculate the rate of change of a function at a particular point in a fixed direction. In this article you will learn more about finding directional derivatives. This guide will help you understand the steps for calculating the directional derivative of a function.

How do you solve for directional derivatives

The directional derivative is basically a derivative that is calculated in a particular direction using a unit vector in that direction. We should have prior knowledge of partial derivatives and gradients to find directional derivatives. These concepts give a better understanding of finding directional derivatives with an angle. Here are some steps that help to find it more efficiently.

Find the unit vector of the given vector v=(a,b).
Find the partial derivative of the given function f(x,y) with respect to x and y.
Take the dot product of the unit vector and the partial derivatives off(x,y).
Or, you can simply use the formula of directional derivative that is
Simplify to get a solution.

Method to solve Directional Derivative

There are two ways to find directional derivatives of any function along a specific direction. These are

By using directional derivative formula.
By taking the dot product of gradient and the unit vector.

Here we are providing you step-by-step solutions of different examples to gain a good understanding. You can also understand the difference between derivatives and directional derivatives.

Directional Derivative Example no.1

For f (x,y) = x²y, find the directional derivative at a point (3,2) in the direction of (2,1).

We can solve this example, either by finding gradients or by using formulas.

Step-1

Let v = 2i + j

Unit vector in the direction of v = 2i + j is,

$$ v\;=\;\frac{v}{|v|}\;=\;\frac{2i+1j}{\sqrt 5}\;=\;\frac{2}{\sqrt 5}i\;+\;;\frac{1}{\sqrt 5}j\;\; ∴ \;\; |v|\;=\;\sqrt{1^2\;+\;2^2}\;=\;\sqrt{5}$$

Step-2

For f (x,y) = x²y, partial derivatives are

$$ \frac{df}{dx}\;=\;2xy$$ $$\frac{df}{dy}\;=\;x^2$$

And the gradient is

$$∆f(x,y)\;=\;2xyi\;+\;x^2j$$

At 3,2,

$$∆f(3,2)\;=\;2(3)(2)i\;+\;3^2j\;=\;12i\;+\;9j$$

Step-3

The derivative of f in the direction of (2,1) at the point (3,2) is

$$D_vf(3,2)\;=\;f_x(3,2)v_1\;+\;f_y(3,2)v_2$$ $$D_vf(3,2)\;=\;\frac{24}{\sqrt 5}\;+\;\frac{9}{\sqrt 5}\;=\;\frac{33}{\sqrt5}$$

Directional derivative Example no. 2

For f (x,y,x) = x²z + y³z²-xyz, find the directional derivative in the direction of v = (-1,0,3).

Step-1

Let v = -1i + 0j + 3k

Unit vector in the direction of v = -i + 3k is,

$$v\;=\;\frac{v}{|v|}\;=\;\frac{-i+3k}{\sqrt 10}\;=\;\frac{-1}{\sqrt 10}i\;+\;\frac{3}{10}k\;\; ∴\;\;|v|\;=\;\sqrt{(-1)^2+3^2}\;=\;\sqrt{10}$$

Step-2

For f(x,y,x) = x²z + y³z²-xyz, partial derivatives are

$$\frac{df}{dx}\;=\;2xz\;-\;yz$$ $$\frac{df}{dy}\;=\;3y^2z^2\;-\;xz$$ $$\frac{df}{dy}\;=\;x^2\;+\;2y^3z\;-\;xy$$

Step-3

The derivative of f in the direction of v = (-1,0,3) is

$$D_vf(x,y,z)\;=\;f_x(x,y,z)v_1\;+\;f_y(x,y,z)v_2\;+\;f_z(x,y,z)v_3$$ $$D_vf(x,y,z)\;=\;(2xz\;-\;yz)(\frac{-1}{\sqrt10})\;+\;(3y^2z^2\;-\;xz)(0)\;+\;(x^2\;+\;2y^3z-xy)(\frac{3}{\sqrt10})$$ $$D_vf(x,y,z)\;=\;(\frac{1}{\sqrt 10})(-2xz\;+\;yz\;+\;3x^2\;+\;6y^3z\;-\;3xy)$$ $$D_vf(x,y,z)\;=\;(\frac{1}{\sqrt 10})(3x^2\;+\;6y^3z\;-\;2xz\;+\;yz\;-\;3xy)$$

How to Find Directional Derivatives Using a Calculator?

We provide you an easy and fast way to calculate directional derivatives of any function using a tool that is a directional derivative calculator. You can use it to make your calculations faster than manual. See the below steps to find derivative with this tool

Find the tool by searching calculatores from your browser and select directional derivative calculator from the section of derivatives.
On the calculator page, enter the function in the “Enter Function” box.
If you want to compute directional derivative for 2D then choose f(x,y) and for 3D choose f(x,y,z).
Now write the values for the vector in U1,U2
Write x and y coordinates.
In the last step, click on the calculate button.

Directional Derivative Real Life Examples

As you have learnt, the directional derivatives indicate that either a function is increasing or decreasing in a particular direction. It has many applications in mathematics and physics as well as in our real life. Let’s discuss some of them.

In mathematics, it is mostly used to find the slope of a surface in three dimensional space.
The use of directional derivatives in modelling infectious disease is a most excellent application. In this model, the possibility of a person to get sick by the Covid-19 is efficiently described.
In the stock market, the variation of price of gold or dollar can be predicted with it.
Direction derivatives also help in machine learning and finding the speed of your car in a specific direction.

It means that this concept has a great impact not only in mathematics and physics, it helps to solve many real life problems.

FAQ’s

What is a Directional Derivative in Calculus?

It is the rate of change of a function at a point in a specific direction. The unit vector is used in that direction to find respective derivatives with an angle. The directional derivative is calculated by taking the dot product of gradient and the unit vector that is

$$D_vf(x,y)\;=\;∇f(x,y).v$$

Is the Directional Derivative the Slope?

Yes, it is a slope of a function at a point in a particular direction. It is a simple concept of directional derivative that is the slope of a function at any point in a specific direction.

Does Directional Derivative imply Continuity?

Yes, it implies continuity along straight lines in various directions. But it does not ensure the continuity of a function.

Is the directional derivative a vector?

No, it is not a vector because it is the dot product of gradient and unit vector at a given point. The dot product always results in a scalar quantity. Therefore, it is not a vector.

How to find Directional Derivative of a Function?

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