# Partial Derivative Calculator

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## Partial derivative calculator

Partial derivative calculator is used to determine the differential of the multivariable functions with respect to any corresponding variable. Our first partial derivative calculator differentiates constant, linear, or polynomial functions several times w.r.t different variables.

## How does this partial differential calculator work?

Follow the below steps to calculate the partial differentiation of any function.

• Input a multivariable function f(x, y)
• Select the variable i.e., x or y.
• Hit the load examples key to use the sample examples.
• Press the calculate button to get the result of the multivariable function with respect to one variable.
• Click the possible intermediate steps to view the step-by-step solution of the given function.
• Hit the clear button to recalculate.

## What is a partial derivative?

A function of multiple variables is the instantaneous rate of change of slope of the function in one of the coordinate directions is known as a partial derivative. It works the same way as a single variable derivative with all other variables treated as constant.

The notation used to find the partial differentiation of multivariable function is $$\frac{\partial }{\partial x}\:or\:\frac{\partial }{\partial y}$$

Equation of f(x, y) with respect to x.

$$\frac{\partial }{\partial x}\left(f\left(x,y\right)\right)$$

Equation of f(x, y) with respect to y.

$$\frac{\partial }{\partial y}\left(f\left(x,y\right)\right)$$

## How to calculate partial derivative?

Following are a few examples of multivariable functions solved by our partial differentiation calculator.

Example 1

Calculate the partial derivative of $$x^2+2xy+z$$ with respect to x.

Solution

Step 1: Use the notation of partial derivative.

$$\frac{\partial }{\partial x}\left(x^2+2xy+z\right)$$

Step 2: Now differentiate the given multivariable function with respect to x.

$$\frac{\partial }{\partial x}\left(x^2+2xy+z\right)=\frac{\partial }{\partial x}\left(x^2\right)+\frac{\partial }{\partial x}\left(2xy\right)+\frac{\partial }{\partial x}\left(z\right)$$

$$\frac{\partial \:}{\partial \:x}\left(x^2+2xy+z\right)=\frac{\partial \:}{\partial \:x}\left(x^2\right)+2y\frac{\partial \:}{\partial \:x}\left(x\right)+\frac{\partial \:}{\partial \:x}\left(z\right)$$

$$\frac{\partial \:}{\partial \:x}\left(x^2+2xy+z\right)=\left(2x\right)+2y\left(1\right)+\left(0\right)$$

$$\frac{\partial \:}{\partial \:x}\left(x^2+2xy+z\right)=2x+2y$$

Step 3: Similarly, the partial differentiation of $$x^2+2xy+z$$ with respect to y.

$$\frac{\partial \:}{\partial y}\left(x^2+2xy+z\right)=2x$$

Example 2

Calculate the partial derivative of $$xsin\left(y\right)+cos\left(x\right)$$ with respect to x & y.

Solution

Step 1: Use the notation of partial derivative.

$$\frac{\partial }{\partial x}\left(xsin\left(y\right)+cos\left(x\right)\right)$$

Step 2: Now differentiate the given multivariable function with respect to x.

$$\frac{\partial }{\partial x}\left(xsin\left(y\right)+cos\left(x\right)\right)=\frac{\partial }{\partial x}\left(xsin\left(y\right)\right)+\frac{\partial }{\partial x}\left(cos\left(x\right)\right)$$

$$\frac{\partial }{\partial x}\left(xsin\left(y\right)+cos\left(x\right)\right)=sin\left(y\right)\frac{\partial }{\partial x}\left(x\right)+\frac{\partial }{\partial x}\left(cos\left(x\right)\right)$$

$$\frac{\partial }{\partial x}\left(xsin\left(y\right)+cos\left(x\right)\right)=sin\left(y\right)\left(1\right)+\left(-sin\left(x\right)\right)$$

$$\frac{\partial }{\partial x}\left(xsin\left(y\right)+cos\left(x\right)\right)=sin\left(y\right)-sin\left(x\right)$$

Step 3: Now differentiate the given multivariable function with respect to y.

$$\frac{\partial }{\partial y}\left(xsin\left(y\right)+cos\left(x\right)\right)=\frac{\partial }{\partial y}\left(xsin\left(y\right)\right)+\frac{\partial }{\partial y}\left(cos\left(x\right)\right)$$

$$\frac{\partial }{\partial y}\left(xsin\left(y\right)+cos\left(x\right)\right)=x\frac{\partial }{\partial y}\left(sin\left(y\right)\right)+\frac{\partial }{\partial y}\left(cos\left(x\right)\right)$$

$$\frac{\partial }{\partial y}\left(xsin\left(y\right)+cos\left(x\right)\right)=x\left(cos\left(y\right)\right)+\left(0\right)$$

$$\frac{\partial }{\partial y}\left(xsin\left(y\right)+cos\left(x\right)\right)=xcos\left(y\right)$$

### References

1. what is Partial derivatives. Brilliant Math & Science Wiki. (n.d.)
2. Examples of partial derivatives. Calculus III - partial derivatives. (n.d.)