Partial Derivatives

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In a beginning calculus course a student learns to define the derivative using limits and then learns multiple rules to learn how to differentiate functions without using limits. In this post I will discuss the concept of a partial derivative and give some examples of how to calculate them.

When taking the derivative of a single variable function such as f(x) there is only one way to do it, by taking the derivative with respect to the variable x. Suppose instead that we have a multivariable function f(x, y). Since this function has two independent variables it is possible to take the derivative of the function with respect to x and with respect to y and these are called partial derivatives.

The partial derivatives can be defined using limits just as the derivative of a single variable function is defined using a limit. In this post we will not go in to the limit definition but instead use intuition to calculate the partial derivatives of a function.

The partial derivatives of f(x, y) are denoted by f_x and f_y and are called the derivatives with respect to x and with respect to y. We can find the partial derivative by treating all the variables, except the one we are differentiating with respect to, as constants and then differentiating like we would a function of a single variable.

For our first example of how to do this consider the function f(x, y) = xy. To find f_x we treat y as a constant and take the derivative normally. If y is a constant then we are essentially just differentiating the function x which we know has derivative equal to 1. Thus it follows that f_x = 1 * y = y. Similarly, if we want to find f_y then we treat x as a constant and differentiate with respect to y to get f_y = x * 1 = x.

For the next example we will find the partial derivatives of f(x, y) = x² + y². For the partial derivative with respect to x we have f_x = 2x + 0 = 2x because we treat y² as a constant. Similarly, we have f_y = 0 + 2y = 2y.

In the above two examples we have calculated the partial derivatives for functions of two variables. Partial derivatives can be calculated for a function of any number of variables and there is one partial derivative for each variable of the function. Let us now calculate the partial derivatives of the function f(x, y, z) = xcos(y) + z. This function has three partial derivatives, one for each variable. We have f_x = 1 * cos(y) + 0 = cos(y), f_y = x(-sin(y)) + 0 = -xsin(y) and f_z = 0 + 1 = 1.

In this post we have shown how to calculate the partial derivatives of multivariable functions and have shown that for each variable of a function there is a partial derivative. The partial derivative is calculated by holding all but one of the variables fixed and then taking the derivative as if it were a function of one variable.

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References:

https://en.wikipedia.org/wiki/Partial_derivative
http://mathworld.wolfram.com/PartialDerivative.html

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All content in this post was created by myself.