Partial Derivative Calculator

Partial Derivative Calculator computes the value of the partial derivatives for a given function. The process of obtaining the partial derivatives of a function is known as partial differentiation.

What is a Partial Derivative Calculator?

Partial Derivative Calculator is an online tool that helps to differentiate a function and obtain its partial derivatives. Vector calculus and differential geometry see the use of partial derivatives. To use this Partial Derivative Calculator, enter the function in the given input box.

Partial Derivative Calculator

How to Use Partial Derivative Calculator?

Please follow the steps given below to find the partial derivatives using the online partial derivative calculator:

• Step 1: Go to Cuemath’s online partial derivative calculator.
• Step 2: Enter the function with respect to x and y in the given input box of the partial derivative calculator.
• Step 3: Click on the "Calculate" button to find the value of the partial derivatives.
• Step 4: Click on the "Reset" button to clear the field and enter new values.

How Does Partial Derivative Calculator Work?

When a function is in terms of one variable only, we can use simple differentiation to find its derivatives. In contrast, we use partial differentiation when the given function is in terms of two or more variables. In partial differentiation, we differentiate the given function with respect to one variable while the other variables are treated as constants. Suppose we have a function that depends on two variables x and y given as f (x, y). The steps to find the partial derivatives of this function are given as follows:

• Differentiate the function with respect to x. Here, the terms containing the y variable will be taken as constants. The partial derivative of a function with respect to x is denoted by \(f_{x}\), \(f'_{x}\), \(\partial _{x}f\) or \(\partial f / \partial x\).
• Now differentiate the function with respect to y. All the terms with the x variable will be treated as constants. This will be denoted as \(f_{y}\), \(f'_{y}\), \(\partial _{y}f\) or \(\partial f / \partial y\).

The formula to find the partial derivatives of a function is given as follows:

Solved Examples on Partial Derivative Calculator

Example 1:

Find the partial derivatives of 5x³ + 2y² and verify them using the partial derivative calculator.

Solution:

Given: f(x,y) = 5x³ + 2y²

Differentiating with respect to x

\(f_{x}\) = \(\frac{\partial }{\partial x} (5x^{3} + 2y^{2})\)

\(f_{x}\) = \(\frac{\partial }{\partial x} 5x^{3}\) + \(\frac{\partial }{\partial x} 2y^{2}\)

\(f_{x}\) = 5 × 3x^{3 - 1} + 0

\(f_{x}\) = 15x²

Differentiating with respect to y

\(f_{y}\) = \(\frac{\partial }{\partial y} (5x^{3} + 2y^{2})\)

\(f_{y}\) = \(\frac{\partial }{\partial y} 5x^{3}\) + \(\frac{\partial }{\partial y} 2y^{2}\)

\(f_{y}\) = 0 + 2 × 2y^{2 - 1}

\(f_{y}\) = 4y

Example 2:

Find the partial derivatives of y sin(x) and verify them using the partial derivative calculator.

Solution:

Given: f(x,y) = y sin(x)

Differentiating with respect to x

\(f_{x}\) = \(\frac{\partial }{\partial x} y sin(x)\)

\(f_{x}\) = y \(\frac{\partial }{\partial x} sin(x)\)

\(f_{x}\) = y cos (x)

Differentiating with respect to y

\(f_{y}\) = \(\frac{\partial }{\partial y} y sin(x)\)

\(f_{y}\) = sin(x) \(\frac{\partial }{\partial y} y \)

\(f_{y}\) = sin(x)

Similarly, you can use the partial derivative calculator to find the partial derivatives for the following:

• x³ / 2y
• ycos(x) + sin(y)

☛ Related Articles:

• Diffferentiation
• Partial Derivative

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