Partial Derivative Calculator





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Introduction to Partial Derivative Calculator

A partial derivative calculator is an online tool for finding the derivatives of multiple variables. This partial differentiation equation calculator is a freely available tool.

partial derivative calculator with steps

A partial differentiation calculator is a tool for understanding the nature of a function depending upon more than a single variable. This multivariable derivative calculator makes it easier for the students and other users to do differentiation quickly.

In this website, we also offer successive differentiation calculator for finding the derivative once, twice, thrice and upto nth times.

What is a Partial Differentiation Calculator?

In Calculus, partial derivatives are the functions having more than one variable in the function, but we differentiate the whole function with respect to one of these variables. This process of calculation is a mind confusing for students due to which we offer this free calculator. The partial derivatives calculator provides the step by step and accurate solution of any type of problem related to partial differentiation.

The partial differentiation calculator is very easy to operate. You can use it right on your device. A partial derivative calculator is a tool for calculating an ordinary derivative function of a variable with a fixed value of the other variable.

The multivariable derivative calculator is an online calculator that is used to find the derivative of different variables. The calculator is free to use have no charges and the user can use it independently and can solve as many problems as he/she can. You can also try our dy dx calculator to find derivatives of implicitly defined functions.

Formula Used by Partial Derivative Calculator with Steps

Parial derivative of any function f(x,y) depends upon the two different variables "x" and "y" at the same time. But for the calculation of partial differentiation of a function, we differentiate the whole function with respect to one of these variables.

Following is the formula used in calculator to get the partial derivative of function f(x,y) w.r.t "x":

$$ \frac{∂f}{∂x} = \frac{∂f}{∂u}\frac{∂u}{∂x} \;+\; \frac{∂f}{∂v}\frac{∂v}{∂x} $$

Similarly, partial derivative of a function, w.r.t "y" is:

$$ \frac{∂f}{∂y} = \frac{∂f}{∂u}\frac{∂u}{∂y} \;+\; \frac{∂f}{∂v}\frac{∂v}{∂y} $$

Let's see how to calculate partial derivative with an example:

Example: Integrate the following function w.r.t "x"

$$ u(x,y) \;=\; \int xy dx \;+\; f(y) $$


f(y)= any differentiable function of y

$$ u(x,y) \;=\; \frac{x^2}{2} \;+\; f(y) $$

By using u(0,y) = 3y

$$ 3y \;=\; u(0,y) \;=\; \frac{x^2}{2} \;+\; f(y) $$ $$ f(y) \;=\; 3y $$ $$ u(x,y) \;=\; \frac{x^2}{2} \;+\; 3y $$

So in the same way, our calculator solve for partial derivatives of the function with detailed solution.

How does Partial Differential Calculator Work?

The user interface of the partial derivative calculator is very easy to use and understand. By following the simple given steps one can get the solution to his/her problems. This calculator usually follows the following steps to calculate the partial derivative of the given function:

  1. First of all, Enter the function in the required input field.
  2. Select the variable w.r.t to which you want to differentiate.
  3. Now click on the "CALCULATE" button to get the partial derivative.
  4. Finally, the derivative of the function having more than one variable will display on the screen. Not it only contains the partial derivative, but also it has possible intermediate steps, plots and trigonometric form of given function.

How to Find the Partial Differential Equation Calculator?

Finding the multivariable derivative calculator online is not like to climb mount everest. The following methods will help you to find the partial derivative calculator with steps:

Method 1: Enter the keywords like "partial derivative calculator","partial differentiation calculator" or "calculate partial derivative" in the search bar of Google, Bing, Yandex or any search engine you prefer to use. Hopefully we will be available on the top results of your search engine.

Method 2: Search the calculator for partial derivative by our direct link i.e.

Also you can visit derivative calculator with steps and search for any of amazing tool related to find derivative of a function.

Benefits of Partial Derivative at a Point Calculator

The partial differential equation calculator with steps has the following benefits:

  • The first-order partial derivative calculator gives you accurate and authentic results.
  • This calculator makes the calculations faster and easier.
  • As like the 2nd derivative calculator, the results of this second order partial derivative calculator are 100% accurate and reliable.
  • This calculator has an easy user-friendly interface for its users.
  • The partial differential calculator with steps gives you step-by-step instructions to solve the problem.
  • It is easier to use and keeps the user away from hectic manual calculations.

Why Chain Rule method is used in Partial Derivatives Calculator?

The partial differentiation calculator also uses the chain rule method for solving the derivatives of the given functions. Due to this, it is also known as a multivariable derivative calculator. The chain rule method is used for finding the derivative of composite functions.

Related: You can also try our multivariable chain rule calculator for online calculation of composite functions.

Frequently Asked Question

Find the First Partial Derivatives of the Function. f(x, y) = ax + by cx + dy

The first partial derivatives of the given function: f(x,y) = ax + by, cx + dy, is

  • For function: f(x,y) = ax + by,

$$ With\; respect\; to\; x\;: \frac{\partial f}{\partial x} \;=\; a $$

$$ With\; respect\; to\; y\;: \frac{\partial f}{\partial y} \;=\; b $$

  • For the function f(x,y) = cx + dy,

$$ With\; respect\; to\; x\;: \frac{\partial f}{\partial x} \;=\; c $$

$$ With\; respect\; to\; y\;: \frac{\partial f}{\partial y} \;=\; d $$

The two variable derivative calculator determines each derivative easily as well.

Find the First Partial Derivatives of the Function. z = x sin(xy)

  • With respect to x:

$$ \frac{\partial z}{\partial x} \;=\; \frac{\partial }{\partial x} \biggr(x sin(xy) \biggr) $$

Using product rule,

$$ 1 . sin(xy) + x. \frac{\partial }{\partial x} \biggr(sin(xy) \biggr) $$

Differentiating sin(xy):

$$ \;=\; sin(xy) + x . cos(xy) . y $$


$$ \frac{\partial z}{\partial x} \;=\; sin(xy) + x cos(xy) y $$

  • With respect to y:

$$ \frac{\partial z}{\partial y} \;=\; \frac{\partial }{\partial y} \biggr( x sin(xy) \biggr) $$

Using the product rule,

$$ \;=\; x . cos(xy) . x $$


$$ \frac{\partial z}{\partial y} \;=\; x^2 cos(xy) $$


$$ \frac{\partial z}{\partial x} \;=\; sin(xy) + x cos(xy) y $$

$$ \frac{\partial z}{\partial y} \;=\; x^2 cos(xy) $$

Find the First Partial Derivatives of the Function. f(x, t) = x ln(t)

  • With respect to x:

$$ \frac{\partial f}{\partial x} \;=\; \frac{\partial }{\partial x} \biggr( x ln(t) \biggr) $$

Using the product rule,

$$ 1. ln(t) + x. \frac{\partial }{\partial x} \biggr(ln (t) \biggr) $$

Differentiating ln(t),

$$ \;=\; ln(t) $$

  • With respect to t:

$$ \frac{\partial f}{\partial t} \;=\; \frac{\partial }{\partial t} \biggr(x ln(t) \biggr) $$

Using the product rule,

$$ \;=\; x . \frac{1}{t} $$


$$ \frac{\partial f}{\partial x} \;=\; ln(t) $$

$$ \frac{\partial f}{\partial t} \;=\; \frac{x}{t} $$

How to Estimate Partial Derivatives from Contour Maps

To estimate the partial derivatives from contour maps, follow some simple steps:

  • Choose the variable and pick two nearby contour lines.
  • Now find how much the value of the function changes in these contour lines.
  • Determine the distance between these contour lines.
  • Now divide the change in function value by the distance and you will get the answer.

How to Take Partial Derivative

Taking the partial derivative means finding the rate at which the function changes with respect to the given variable. For taking the partial derivative: 

  • Identify the function and choose the variable
  • Take other variables as constant and simplify the result.