Partial Derivative Calculator

Partial Derivative Calculator:

This partial derivative calculator differentiates the multivariable functions step-by-step with respect to your chosen variable and considers the other variables as constants.

What is the Partial Derivative of a Function?

"The partial derivative is defined as the derivative of a multivariable function with respect to one of its variables, keeping all the variables as constants."

It measures how a function changes when one of its variables changes, treating the other variables as constants.

The symbol "∂/∂" is used to indicate partial differentiation.

Understanding Partial Derivatives:

Consider a function with two independent variables, f(x, y):

If you need to find the partial derivative of the function with respect to “x”, then you should keep the variable “y” constant and differentiate with respect to “x”. The notation for this is ∂x/∂f or fₓ.
On the other hand, if you need to differentiate the function with respect to “y”, then make the variable “x” constant and differentiate with respect to the variable “y”. The notation for this is ∂y/∂f or fᵧ

Partial Derivative Formula:

To find the partial derivative of a function with respect to “x”:

∂f/∂x = lim_h→0 [ (f(x + h, y) - f(x, y)) / h ]

To find the partial derivative of a function with respect to “y”:

∂f/∂x = lim_h→0 [ (f(x, y + h) - f(x, y)) / h ]

How Do You Calculate A Partial Derivative?

Step #1: Determine the function f(x,y,z,...) and the variable you want to differentiate with respect to (e.g., x, y, z)
Step #2: Keep all the other variables as constants. Treat them as fixed numbers while performing the differentiation process
Step #3: Now, implement the standard rules of differentiation (power rule, product rule, chain rule, etc.) according to the selected variable

Note: All terms that do not contain the chosen variable will be considered as a constant, and their derivative will be zero.

Second Partial Derivatives:

Second order partial derivatives are found by taking the partial derivative of a first-order partial derivative. For a function f(x,y):

fxx or ∂²x/∂f² = ∂/∂x(∂f/∂x): The partial derivative of the first partial derivative of f with respect to x, again with respect to x
fyy or ∂²f/∂y² = ∂/∂y(∂f/∂y): The partial derivative of the first partial derivative of f with respect to y, again with respect to y

Example of Partial Derivative:

Find partial derivative of the function f(x, y) = 2x² + eʸ - 3xy² with respect to x, denoted as ∂f/∂x.
Solution:

Step #1: Identify the function and the variable of differentiation
We are given the function f(x,y)= 2x²+eʸ−3xy² and we want to differentiate with respect to x.

Step #2: Consider y as a constant
This means that terms involving only y (like ey) will have a derivative of zero with respect to x.

Step #3: Apply standard differentiation rules

Using the power rule (dxd(xn)=nxn−1), the derivative of 2x² is 2×2x²⁻¹ = 4x¹ = 4x.

Since we are keeping y as a constant, the derivative of eʸ with respect to x is 0

Here, -3y² is treated as a constant coefficient. The derivative of x with respect to x is 1. So, the derivative of - 3xy² with respect to x is - 3y²×1 = −3y²

Step #4: Add the derivatives of each term
We get: ∂f/∂x = 4x + 0 - 3y²

Step #5: Simplify the result: ∂f/∂x = 4x - 3y²

Therefore, the partial derivative of f(x, y) = 2x²+eʸ−3xy² with respect to x is 4x - 3y².

To verify your work or to quickly compute partial derivatives for multivariable functions, use our multivariable partial derivative calculator. It provides detailed, step-by-step solutions to help you understand the process.

What Are the Rules of Partial Derivatives?

1. Product Rule:

If u = f(x, y).g(x, y):

u_x = ∂u/∂x = g(x, y) * ∂f/∂x + f(x, y) * ∂g/∂x

And, u_y = ∂u/∂y = g(x, y) * ∂f/∂y + f(x, y) * ∂g/∂y

2. Quotient Rule:

u = f(x, y)/g(x, y), where g(x,y) ≠ 0:

u_x = (g(x, y) * ∂f/∂x - f(x, y) * ∂g/∂x) / [g(x, y)]²

And, u_y = (g(x, y) * ∂f/∂y - f(x, y) * ∂g/∂y) / [g(x, y)]²

3. Power Rule:

u_x = n |f(x, y)|^n-1 ∂f/∂x

And, u_y = n |f(x, y)|^n-1 ∂f/∂y

4. Chain Rule:

In the chain rule, each variable of a multivariable function is the function of another variable. Therefore, the derivative of the function is the sum of the chain rule, applied to all the variables.

Chain Rule for One Independent Variable:
If z = f(x(t), y(t)):

∂z/∂t = ∂z/∂x ⋅ ∂x/∂t + ∂z/∂y ⋅ ∂y/∂t

Chain Rule for Two Independent Variables:
If z = f(x(u,v), y(u,v)):

∂z/∂u = ∂z/∂x ⋅ ∂x/∂u + ∂z/∂y ⋅ ∂y/∂u

and

∂z/∂v = ∂z/∂x ⋅ ∂x/∂v + ∂z/∂y ⋅ ∂y/∂v

Partial Derivative of Natural Logarithm (In):

Perform the same procedure as you do for the normal derivative(apply the chain rule, product rule, quotient rule, and the specific rule for ln(u))
Keep all the other variables as constants
Perform the partial derivative for all the variables involved in the function

How to Use the Partial Derivative Calculator?

Follow these steps to correctly use our online partial derivative calculator to differentiate the given functions:

Enter a function for differentiation
Select the variable for which you want to differentiate the function from the drop-down menu
Click on the “Calculate” button and get the partial derivative of a function with step-by-step calculations

Applications of Partial Derivatives:

Partial derivatives have a great use in various fields of life, including:

Physics: In this field, the partial derivative is widely used to form the basis of fundamental laws of nature like Maxwell’s equations, Schrödinger's equation, Newton's laws, etc
Economics: The partial derivatives play an important role in the study of the production functions, utility functions, and cost functions, etc
Computer Science: They help to optimize the algorithms that are used by machine learning and artificial intelligence(AI)
Medicine: The partial derivatives help to regenerate the images from the given raw data. It is beneficial for MRI and CT scans
Environmental Science: Partial derivatives are also used in the study of population dynamics, pollution, and other environmental factors. They help to solve differential equations

FAQ’s:

Can this Calculator Compute Partial Derivatives with Respect to Multiple Variables?

Yes, our calculator can handle functions containing multiple variables. When you input your function, you can specify which variable you want to differentiate with respect to at that time.

What Is The Difference Between a Regular Derivative and a Partial Derivative?

The main difference between a regular derivative and a partial derivative lies in the function upon which they operate. A regular derivative differentiates functions that contain only one independent variable. In contrast, partial derivatives are used to differentiate multivariable functions, but the differentiation is performed with respect to one specific variable at a time, keeping all the other variables as constants.

References:
From the source of Wikipedia.org: Partial Derivative.

From the source of KhanAcademy.org: Introduction to Partial Derivatives.