Introduction to Directional Derivative Calculator:
Directional derivative calculator angle is the best online source that helps you to find the directional derivative of a given function in multidimensional space. Our tool determines the rate of a change of function at a specific point.
The directional derivative at a point calculator is a handy tool for students, teachers, professionals, or researchers who want to get a solution of multivariable function in a specific direction of scalar field.
What is a Directional Derivative?
The directional derivative is a process that is used to find the multivariable function f(x, y, z,…) at a point a = (a1, a2, a3,…) to measure the rate at which the given function changes, going towards a in the direction of a vector u = (u1, u2, u3,…).
The directional derivative process is used for partial derivative functions where you can find the scalar function in a particular direction. As it determines the maximum or minimum value of several functions along the given direction that describes function behavior in a scalar field.
Formula of Directional Derivative:
The directional derivative is mathematically defined as a multivariable function f, of a given function at point a in the direction of vector u. The directional derivative formula used by the directional derivative calculator with angle is given as:
$$ D_u f(a) \;=\; \nabla f(a) . u $$
- Du: the directional derivative toward the u vector
- f(a): function at a point a
- ▽: the gradient of a multivariable function
- Here, the gradient ▽ of a function f at a point a is f(a).
$$ \nabla f(a) \;=\; \left( \frac{\partial f}{\partial x_1}(a),\; \frac{\partial f}{\partial x_2}(a),\; \frac{\partial f}{\partial x_3}(a), … \right) $$
How to Calculate Directional Derivative?
The directional derivative online calculator is an easy way to calculate the partial derivation in a specific direction of scalar field. The process of direction derivative calculation is given as:
Step 1:
Identify the given function f(x, y, z…) and a specific point value a(a1, a2,..) along the u direction.
Step 2:
Evaluate the gradient ▽ of a given function as it takes the partial differential with respect to its variable along the direction.
Step 3:
After differentiation, simplify the expression and then apply the value of point a(a1, a2,..)
Step 4:
After putting the point value in the differentiation function simplify it to get the solution.
Solved Example of Directional Derivative:
The directional derivative calculator angle solves the directional derivative problem without any hurdle. So, here’s an example to let you know how it actually calculates,
Example:
Determine the directional derivative of the following,
$$ f(x,y) \;=\; x^2 - xy + 3y^2\; in\; the\; direction\; of\; \vec{u} \;=\; (cos\; \theta) \hat{i} + (sin\; \theta) \hat{j} $$
Solution:
Identify the given function f(x, y, z…) at a particular point and direction,
$$ f(x,y) \;=\; x^2 - xy + 3y^2 $$
$$ \vec{u} \;=\; (cos\; \theta) \hat{i} + (sin\; \theta) \hat{j} $$
Compute the gradient of the function:
$$ D_{\vec{u}} f(x, y, z) \;=\; f_x (x,y) cos\; \theta + f_y (x,y) sin\; \theta $$
$$ \;=\; (2x - y) \frac{3}{5} + (-x + 6y) \frac{4}{5} $$
$$ =\; \frac{6x}{5} - \frac{3y}{5} - \frac{4x}{5} + \frac{24y}{5} $$
$$ =\; \frac{2x + 21y}{5} $$
Put the value of given point a in the above function and simply use the expression to get the solution,
$$ D_{\vec{u}} f(1,2) \;=\; \frac{2(-1) + 21(2)}{5} \;=\; \frac{-2 + 42}{5} \;=\; 8 $$
How to Use Directional Derivative Calculator?
The directional derivative at a point calculator has a simple design that makes it easy for you to use it for the evaluation of directional derivative problems. Follow some simple steps that are given as:
- Enter the given function in the given input field that you want to differentiate with a specific point.
- Put the value of the direction of a vector u(u1, u2,..) in the next input field.
- Put the value of the point around which the directional derivative is evaluated.
- Check the given directional derivative function value before clicking the calculate button of the directional derivative online calculator.
- Click the “Calculate” button to get the result of your directional derivative problem.
- If you are trying our tool for the first time then you can use the load example option to learn more about this process.
- The “Recalculate” button brings you back to the home page where you can find more solutions of directional derivative problems.
Results from Directional Derivative at a Point Calculator:
The directional derivative calculator with angle gives you the solution from a given function when you give it an input. It includes as:
- Result Option:
When you click on the result option, it gives you a solution of partial differential function in a specific point.
- Possible Steps:
When you click on it, this option will provide you step by step solution where you get the function in the direction of u.
- Plot Option:
It will give you a graphical representation of directional derivative function to get a better understanding of the result.
Advantages of Directional Derivative Online Calculator:
The directional derivative at a point calculator provides tons of advantages and helps you to calculate the directional derivative of a function. These advantages are:
- It is a free-of-cost tool so you can use it for free to find directional derivative problems with solutions.
- It gives you conceptual clarity for directional derivative process when you use it to solve multiple examples.
- It saves the time and effort that you consume when calculating complex directional derivative functions to find the maximum or minimum value manually.
- It is an adaptable tool that provides you accurate solutions whenever you use it to calculate directional derivatives.
- Our directional derivative calculator angle enables you to use it multiple times for the evaluation of directional derivative problems.
- It is a handy tool so you can access it through an online platform from anywhere.
