Calculus Calculators

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# Partial Derivative Calculator

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## Partial Derivative Lesson

#### Lesson Contents

### How to Calculate a Partial Derivative

Finding the partial derivative of a function by hand is very easy if you already know how to do a normal derivative. When calculating a partial derivative, we are dealing with a function of two or more independent variables. For example, the value of the function f(x, y)= x + y is dependent on the independent variables *x* and *y*, and therefore is a two-variable function.

The partial derivative of this function with respect to *x* is notated as ^{∂f}⁄_{∂x}f(x, y) where *∂* is the partial derivative, *f* is the function, and *x* is the variable it’s in respect to. It is also acceptable to leave out the *f* and write the notation as ^{∂}⁄_{∂x}.

**When calculating a partial derivative with respect to a variable, simply differentiate with respect to that variable, treating the other independent variables as constants.** For example, when calculating ^{∂f}⁄_{∂x}(yx^{2}), we differentiate with respect to *x* and treat *y* as if it were a constant. This results in ^{∂f}⁄_{∂x}(yx^{2}) = 2yx.

### Example Problem

Calculate the partial derivative ^{∂f}⁄_{∂y} of the function f(x, y) = sin(x) + 3y.

Solution:

1.) Since we are differentiating with respect to *y*, we can treat variables other than *y* as constants. So, we will treat *x* as a constant.

2.) The sin(x) term is therefore a constant value. Since differentiating a constant results in zero, sin(x) becomes 0 and is eliminated from the expression.

3.) Applying the derivative power rule to 3y results in:

(1)3y^{(1 – 1)} = 3

4.) Therefore, the partial derivative with respect to *y* is ** ^{∂f}⁄_{∂y}[sin(x) + 3y] = 3**.

## How the Calculator Works

The partial derivative calculator on this page computes the partial derivative of your inputted function symbolically with a computer algebra system, all behind the scenes. The computer algebra system is very powerful software that can logically digest an equation and apply every existing derivative rule to it in order. It follows the same steps that a human would when calculating the derivative.

A symbolic derivative is done using algebra and derivative rules which allows it to maintain the function’s variables and values perfectly. The word *symbolic* is used because the numbers and variables are treated as symbols rather than approximated numbers that get rounded by the computer.