Directional Derivative Calculator

directional derivative calculator




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    Introduction to Directional Derivative Calculator?

    A directional derivative calculator is an online tool for finding the derivative at specific points of a vector. This is a freely available tool for finding vector derivatives. It shows you step-by-step solutions to calculate derivatives on implicit functions.

    directional derivative calculator with steps

    What are Directional Derivatives?

    Directional derivative is a concept that helps to understand how a function changes in a particular direction at a specific point. It helps to measure the rate of change of a function in multivariable calculus where function have multiple input variables.

    Directional derivatives are useful in various fields i.e. physics, maths as well as in engineering. It provides a reasoning that how a function changes with respect to specific directions is essential in analysis and graphing.

    Formula Used in Directional Vector Calculator to Perform Calculations

    The formula to calculate the directional derivative is:

    $$ D_v \; f(a,b) \;=\; ∇f(a,b)⋅ v $$

    Where, ∇f(a,b) is the gradient of f at (a,b), given by:

    $$ ∇f(a,b) \;=\; ( \frac{∂x}{∂f} , \frac{∂y}{∂f}) $$

    Note: The dot product in the formula represents the cosine of the angle between the gradient vector and the unit vector v.

    How Does the Directional Derivative Calculator Work?

    To find the Derivative of a vector, the following steps are to be performed:

    1. Open the calculator page, and load the function here in the required field.
    2. You can also experience this tool with an example from our "LOAD EXAMPLE" button.
    3. Enter all the values in the required input fields. Like variables (select f(x, y) or f(x, y, z)), and other additional values (values of U1 and U2) for calculating the value.
    4. Select any number in the “TIME” field that asks you how many times you want to differentiate the function.
    5. Now just click on the "CALCULATE" button.
    6. The Directional Derivative Calculator calculates the two vectors for the given derivative function.

    Finally, the results will display on your screen. It takes the vector derivative of each derivative function that is entered. Now refresh your directional derivatives calculator page for new calculations.

    How to Calculate the Directional Derivative Online?

    To calculate directional derivative of a function f(x,y) at a point (a,b) in the direction of a unit vector v = ⟨v1 , v2⟩, you can follow these steps

    In the first step of calculating directional derivative, you need to find the gradient of your function f(x,y) at the point (a,b), denoted as ∇f(a,b). The gradient of a function may evaluate using the following formula:

    $$ ∇f(a,b) \;=\; ( \frac{∂x}{∂f} , \frac{∂y}{∂f}) $$

    Find ∂x/∂f and ∂y/∂f, and then evaluate them at a point (a,b).

    Secondly, you have to normalize the direction vector "v" to check either it is a unit vector or not. Each vector will get divided by its magnitude to normalize a vector i.e.

    $$ v_{unit} \;=\; \frac{v}{||v||} \;=\; \frac{⟨v_1,v_2⟩}{\sqrt{(v_1)^2+(v_2)^2}} $$

    Later on, go to calculate the dot product. The dot product may calculate for normalized direction vector vunit and the gradient ∇f(a,b) using the below formula:

    $$ ∇f(a,b) \cdot v_{unit} \;=\; \left( \frac{∂x}{∂f} , \frac{∂y}{∂f} \right) \cdot \frac{⟨v_1,v_2⟩}{\sqrt{(v_1)^2+(v_2)^2}} $$

    Now, evaluate this dot product to take it to the simplest level. This will be your directional derivative of a function.

    How to Find the Directional Derivatives Calculator?

    The following are the steps to be followed for finding the tool to calculate the directional derivative:

    First of all, enter the keywords in the search bar. Google shows you some suggestions for your searched calculators.

    Select the calculator directional derivative from Google suggestions to start calculating the derivative of a vector function for the given differential functions.

    After choosing the directional vector calculator with steps, put your given functions in the required fields and calculate your results.

    Benefits of using Directional Derivative at a Point Calculator

    The directional vector calculator are specially designed for students, researchers and all other related people for calculating directional derivative with multiple more benefits. Following is the list of some perks of using this tool for calculate directional derivative:

    • The Directional Derivative Calculator is also for finding the graphs and slope in a three-dimensional direction.
    • It makes accurate, reliable calculations without any errors.
    • It improves the analytical reasoning features of the user.
    • The user interface of this directional vector calculator is very friendly.
    • The directional derivatives calculator shows results in the form of graphs and slopes.
    • This freely available online calculator gives you results in a fraction of a second.
    • The calculator directional derivative gives you the solution with step-by-step instructions.
    • The user doesn’t need to do long, hectic calculations to find the functions' values.
    • This online tool has unlimited access, the user can practice as many examples as he/she can.

    So this is all bout our directional derivative at a point calculator. Explore the wide range of derivative calculus tools with us. We at Derivative Calculators continusely worl to provide you best ever tools for smart and hessel free learning.

    Frequently Asked Question

    How to Find the Maximum Value of a Directional Derivative

    To find the maximum value of the directional derivative you can follow some simple steps,

    • Determine the gradient of the function.
    • Find the unit vector which represents the direction where you want to determine the directional derivative.
    • Now take the dot product of the unit vector and gradient vector
    • Set the derivative equal to zero and solve for critical points to determine the maximum value of the directional derivative
    • Make sure that the direction you are considering does not violate any boundary constraints.
    • Analyze your results now.

    How to Find a Directional Derivative

    To find the directional derivative of a function in a direction of a unit vector on a point, the following formula could be used,

    $$ D_u f(x_0, y_0, z_0) \;=\; \nabla (x_0, y_0, z_0). u $$

    • Determine the gradient vector of the function and find the partial derivatives with respect to x, y, and z.
    • Make sure that the direction vector is a unit vector by dividing the direction vector by it magnitude. If it is a unit vector then its magnitude would be 1.
    • Calculate the dot product of the gradient vector which will give the directional derivative.
    • Now substitute the unit direction vector and the values of gradient vector into the dot product formula and determine the numerical value of the directional derivative.

    What is a Vector Derivative

    The vector derivative is used to calculate vector to represent a vector containing the partial derivatives of a scalar function with respect to the variables. It is also known as the nabla or gradient operator.

    The gradient or nabla operator represents the direction and the rate of change of the scalar function in a space spanned by its variables. Whereas the magnitude represents the rate of maximum change of the function on a point. The vector derivative can be determined using the following formula,

    $$ \nabla f \;=\; \biggr( \frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, … , \frac{\partial f}{\partial x_n} \biggr) $$

    Find the Directional Derivative of f(x y z) at the Point in the Direction of the Vector

    To find the directional derivative of the scalar function, the following formula can be used,

    $$ D_v f(x_0, y_0, z_0) \;=\; \nabla f(x_0, y_0, z_0). \frac{v}{