So far, we've looked at functions with a single input, like $f(x)$. The real world, however, is rarely so simple. A company's profit might depend on production costs and marketing spend. The temperature in a room depends on your location in it, maybe as coordinates $(x, y, z)$. These are multivariable functions.

This raises a question: how do we measure the rate of change for a function with multiple inputs? The regular derivative won't work because it's designed for a single variable. Which variable should we focus on?

The answer is to look at them one at a time. Imagine you're adjusting an audio equalizer. To understand the effect of the bass, you change the bass slider while keeping the treble and midrange sliders fixed. You're isolating one variable to see its impact. Partial derivatives apply this same logic to mathematics.

A partial derivative measures the rate of change of a multivariable function with respect to one variable, while holding all other variables constant.

Let's say we have a function $f(x, y)$. The partial derivative of $f$ with respect to $x$ is written as $\frac{\partial f}{\partial x}$ or $f_x$. To calculate it, you simply treat $y$ as if it were a constant number, like 5 or -10, and then take the derivative with respect to $x$ as you normally would. The same process applies for finding the partial derivative with respect to $y$, $\frac{\partial f}{\partial y}$, where you treat $x$ as a constant.

Let's try an example with the function $f(x, y) = x^2 + 3xy + y^2$. First, we'll find the partial derivative with respect to $x$.

We differentiate term by term, treating $y$ as a constant.

1. The derivative of $x^2$ with respect to $x$ is $2x$.
2. For $3xy$, we treat $3y$ as a constant coefficient of $x$. The derivative is just $3y$.
3. For $y^2$, since $y$ is treated as a constant, $y^2$ is also a constant. The derivative of a constant is 0.

Putting it together, we get:

Now, let's find the partial derivative with respect to $y$, treating $x$ as a constant.

1. The derivative of $x^2$ with respect to $y$ is 0, since $x$ is treated as a constant.
2. For $3xy$, we treat $3x$ as a constant coefficient of $y$. The derivative is $3x$.
3. The derivative of $y^2$ with respect to $y$ is $2y$.

So, the result is:

You might remember that the derivative of a single-variable function, $f'(x)$, gives the slope of the line tangent to the curve at point $x$. A partial derivative does something similar, but in three dimensions.

The graph of a function with two variables, $z = f(x, y)$, is a surface. Imagine this surface floating in a room. To find the partial derivative with respect to $x$ at a point $(x_0, y_0)$, we slice the surface with a vertical plane where $y$ is held constant at $y_0$. This slice creates a curve on the surface.

The partial derivative $\frac{\partial f}{\partial x}$ at $(x_0, y_0)$ is the slope of the line tangent to that specific curve at that point. It tells us how steep the surface is as we move purely in the x-direction.

Likewise, the partial derivative $\frac{\partial f}{\partial y}$ is the slope of the tangent line to the curve created by slicing the surface with a plane where $x$ is held constant. It tells you the steepness as you move purely in the y-direction.

These partial derivatives give us a way to analyze the local behavior of a complex surface by breaking it down into simple, one-dimensional slopes.

Ready to test your understanding?

Understanding how to isolate variables and calculate these slopes is a key first step in mastering the calculus of multiple variables.