Introduction
In calculus, we will often have to work with functions not in one variable, but in two or more. Functions of this type are called $\textbf{multivariable functions}$. Their notation is slightly different. Examples of multivariable functions are $f(x, y) = xy + 1$ and $f(x, y, z) = xy + yz$. The derivative of a multivariable function is referred to as a partial derivative.
Partial Derivatives
When we take derivatives of functions like $f(x) = x^2 + 1$, we often say “take the derivative of $f(x)$ \emph{with respect to $x$}.” Saying “with respect to $x$” means that we’re measuring how $f(x)$ changes as $x$ changes. Similarly, a partial derivative is the derivative of a multivariable function with respect to one variable while keeping the other variables fixed.
Partial derivatives can be written in several ways. Taking the partial derivative of a multivariable $f$ with respect to $x$ is denoted as $$\dfrac{\partial f}{\partial x} = \dfrac{\partial}{\partial x}f(x, y, \dots).$$ Sometimes, the notation $f_x$ is also used to denote the partial derivative of $f$ with respect to $x$.
Since we are working with multivariable functions, we may also have to find the partial derivative of $f$ with respect to other variables, like $y$. In this case, we use $\dfrac{\partial f}{\partial y}$ or $f_y$.
Examples
Let’s try an example. Suppose we want to take the partial derivative of $f(x, y) = xy + x$ with respect to $x$.
To find the partial derivative, we apply the same rules as in single-variable differentiation, but we treat every variable other than $x$ as constants. As a result,
$$ \dfrac{\partial f}{\partial x} = \dfrac{\partial}{\partial x}(xy + x) = \boxed{y + 1}. $$
Let’s try another example. Suppose we want to take the partial derivative of $f(x, y, z) = xy^2 + y^2z$ with respect to $y$. We apply the same idea – treat every variable other than $y$ as constants, and differentiate like in single-variable functions. Using the Power Rule, we have:
\[\begin{align*} \dfrac{\partial f}{\partial y} &= \dfrac{\partial}{\partial y}(xy^2 + y^2z)\\ &= x(2y) + (2y)z\\ &= \boxed{2xy + 2yz}. \end{align*}\]
Understanding partial derivatives is an important skill used in many calculus problems. The underlying idea is similar to the core of implicit differentiation. For more information on derivatives, check out other blogs at iacedcalculus.com.
