Partial Derivative Calculator

Partial Derivative Calculator 

Partial Derivative Calculator is an advanced mathematical tool designed to assist you in finding the partial derivatives of multivariable functions. This tool allows differentiation up to 5 times. 

With a user-friendly interface, built-in keyboard, and easy-to-follow instructions, this tool makes it effortless for users to input the required parameters and receive detailed step-by-step solutions.

How to use this tool?

To use the partial differentiation calculator, follow the given instructions.

• Input the function: Enter the multivariable function for which you want to find the partial derivative in the designated input field.
• Select the variable: Choose the variable with respect to which you want to compute the partial derivative from the dropdown menu or enter it manually in the provided field.
• Specify the number of derivatives: Indicate the order of the derivative (n) by inputting the desired value in the corresponding field. For example, to find the second partial derivative, enter "2".
• Calculate and Review the results: Click on the “Calculate” option and analyze the step-by-step solution provided by the partial differential calculator.

What is partial differentiation?

Partial derivatives in calculus deal with functions of multiple variables. 

The partial derivative of a function with respect to a specific variable measure the rate at which the function changes as that variable changes while keeping all other variables constant. 

It is denoted as ∂f/∂x, where f is the function and x is the variable with respect to which the derivative is taken. The notation ∂ is read as “del”. It distinguishes partial differentiation from simple differentiation.  

Partial derivatives with graphs:

Simple or mono-variable derivatives can be understood with the help of a two-dimensional cartesian plane. 

In partial differentiation, the derivative function involves two or more variables so the plane has to extend to more dimensions as per requirement. 

Imagine a bi-variable (variable a and b) function. For its graph, specify the x-axis and y-axis for input variables (a) and (b) and the z-axis for the output function (i.e derivative).

Now imagine a small change in either one variable (say a) while keeping the other variable (b) constant. Depending on the variable’s value in the function, the graph will change on the z-axis. While it will remain fixed on its position regarding the y-axis (for variable b).

You can simply plot the changing variable and the output only. It is because the remaining variables are constant so they will be useless to plot to determine the rate of change.

Visualizing partial derivatives:

Imagine standing on the surface at a specific point (x, y) and looking in the positive x-direction. The slope of the surface in that direction represents the partial derivative ∂f/∂x.

If the slope is positive, it means that the function is increasing in the x-direction, and if it is negative, the function is decreasing. A slope of zero indicates that the function is not changing in the x-direction at that point.

Similarly, when we take the partial derivative with respect to y, we are evaluating the rate of change of the function as we move along the y-axis while keeping x constant. The slope of the surface in the positive y-direction represents the partial derivative ∂f/∂y.

How to find the partial derivatives?

Identify the function and the variable with respect to which you want to find the partial derivative. Treat all other variables in the function as constants.

Apply the basic rules of differentiation, such as the power rule, product rule, quotient rule, and chain rule, as needed, while differentiating with respect to the chosen variable.

Simplify the resulting expression to obtain the partial derivative. If you need to find higher-order partial derivatives, repeat steps 2 to 4, taking the derivative of the previously obtained partial derivative with respect to the chosen variable.

Example 1:

Let's find the first partial derivative of the function f(x, y) = x²+ 3xy + y³ with respect to x.

Solution

To find the partial derivative of f(x, y) with respect to x, we differentiate the function with respect to x while treating y as a constant:

∂f/∂x = ∂(x² + 3xy + y³)/∂x

Using the power rule and treating y as a constant:

∂f/∂x = 2x + 3y

So, the first partial derivative of f(x, y) with respect to x is 2x + 3y.

Now, let's find the second partial derivative of the function f(x, y) = x² + 3xy + y³ with respect to x.

To find the second partial derivative, we differentiate the first partial derivative with respect to x:

∂²f/∂x² = ∂(2x + 3y)/∂x

Again, treating y as a constant:

∂²f/∂x² = 2

So, the second partial derivative of f(x, y) with respect to x is 2.