What is the partial derivative, how do you compute it, and what does it mean.

What we're building to

  • For a multivariable function, like f, left parenthesis, x, comma, y, right parenthesis, equals, x, start superscript, 2, end superscript, y, computing partial derivatives looks something like this:
  • This swirly-d symbol, \partial , called "del", is used to distinguish partial derivatives from ordinary single-variable derivatives. Or, should I say ... to differentiate them.
  • The reason for a new type of derivative is that when the input of a function is made up of multiple variables, we want to see how the function changes as we let just one of those variables change while holding all the others constant.
  • With respect to three-dimensional graphs, you can picture the partial derivative fx\dfrac{\partial f}{\partial x} by slicing the graph of f with a plane representing a constant y-value and measuring the slope of the resulting curve along the cut.
Intersecting y=0 plane with the graph

What is a partial derivative?

We'll assume you are familiar with the ordinary derivative start fraction, d, f, divided by, d, x, end fraction from single variable calculus. I actually quite like this notation for the derivative, because you can interpret it as follows:
  • Interpret d, x as "a very tiny change in x".
  • Interpret d, f as "a very tiny change in the output of f", where it is understood that this tiny change is whatever results from the tiny change d, x to the input.
In fact, I think this intuitive feel for the symbol start fraction, d, f, divided by, d, x, end fraction is one of the most useful takeaways from single-variable calculus, and when you really start feeling it in your bones, most of the concepts around derivatives start to click.
For example, when you apply it to the graph of f, you can interpret this "ratio" start fraction, d, f, divided by, d, x, end fraction as the rise-over-run slope of the graph of f, which depends on the point where you started.
Interpretation of in a single variable function.

How does this work for multivariable functions?

Consider some function with a two-dimensional input and a one-dimensional output.
f, left parenthesis, x, comma, y, right parenthesis, equals, x, start superscript, 2, end superscript, minus, 2, x, y
There's nothing stopping us from writing the same expression, start fraction, d, f, divided by, d, x, end fraction, and interpreting it the same way:
  • d, x can still represent a tiny change in the variable x, which is now just one component of our input.
  • d, f can still represent the resulting change to the output of the function f, left parenthesis, x, comma, y, right parenthesis.
However, this ignores the fact that there is another input variable y. The input space now has multiple dimensions, so we can change the input in many directions other than the x-direction. For example, what about changing y slightly by some small value d, y? Now if we re-interpret d, f to represent the tiny change to the function that this d, y shift brings about, we would have a different derivative start fraction, d, f, divided by, d, y, end fraction.
Indication that the input of a multivariable function can change in many directions.
Neither one of these derivatives tells the full story of how our function f, left parenthesis, x, comma, y, right parenthesis changes when its input changes slightly, so we call them partial derivatives. To emphasize the difference, we no longer use the letter d to indicate tiny changes, but instead introduce a newfangled symbol \partial to do the trick, writing each partial derivative as fx\dfrac{\partial f}{\partial x}, fy\dfrac{\partial f}{\partial y}, etc.
You read the symbol fx\dfrac{\partial f}{\partial x} out loud by saying "the partial derivative of f with respect to x".

Example: Computing a partial derivative

Consider this function:
f, left parenthesis, start color blueE, x, end color blueE, comma, start color redE, y, end color redE, right parenthesis, equals, start color blueE, x, end color blueE, start superscript, 2, end superscript, start color redE, y, end color redE, start superscript, 3, end superscript
Suppose I asked you to evaluate fx\dfrac{\partial f}{\blueE{\partial x}}, the partial derivative with respect to x, at the input left parenthesis, start color blueE, 3, end color blueE, comma, start color redE, 2, end color redE, right parenthesis.
"What? But I haven't learned how yet!"
Don't worry, it's mostly just the same mechanics as an ordinary derivative.
From the introduction above, you should know that this is asking about the rate at which the output of f changes as we nudge the x-component of the input slightly, perhaps moving from left parenthesis, start color blueE, 3, end color blueE, comma, start color redE, 2, end color redE, right parenthesis to left parenthesis, start color blueE, 3, point, 01, end color blueE, comma, start color redE, 2, end color redE, right parenthesis.
Since we only care about movement in the start color blueE, x, end color blueE-direction, we might as well treat the start color redE, y, end color redE-value as a constant. In fact, we can just plug in start color redE, y, equals, 2, end color redE ahead of time before computing any derivatives:
f, left parenthesis, start color blueE, x, end color blueE, comma, start color redE, 2, end color redE, right parenthesis, equals, start color blueE, x, end color blueE, start superscript, 2, end superscript, left parenthesis, start color redE, 2, end color redE, right parenthesis, start superscript, 3, end superscript, equals, 8, start color blueE, x, end color blueE, start superscript, 2, end superscript
Now, asking how f changes in response to a small shift in start color blueE, x, end color blueE is just an ordinary, single-variable derivative.
Concept check: What is the derivative of this function f, left parenthesis, start color blueE, x, end color blueE, comma, start color redE, 2, end color redE, right parenthesis, equals, 8, start color blueE, x, end color blueE, start superscript, 2, end superscript evaluated at start color blueE, x, equals, 3, end color blueE?

start fraction, d, divided by, start color blueE, d, x, end color blueE, end fraction, f, left parenthesis, start color blueE, x, end color blueE, comma, 2, right parenthesis, equals, start fraction, d, divided by, start color blueE, d, x, end color blueE, end fraction, left parenthesis, 8, start color blueE, x, end color blueE, start superscript, 2, end superscript, right parenthesis, equals, 16, start color blueE, x, end color blueE
Plugging in start color blueE, x, equals, 3, end color blueE, we see the answer must be 16, left parenthesis, start color blueE, 3, end color blueE, right parenthesis, equals, 48.

Without pre-evaluating y

Now suppose I asked you to find fx\dfrac{\partial f}{\blueE{\partial x}}, but I didn't ask you to evaluate it at a specific point. In other words, you should give me new multivariable function which takes any point left parenthesis, start color blueE, x, end color blueE, comma, start color redE, y, end color redE, right parenthesis as its input and tells me what the rate of change of f near that point is as we move purely in the start color blueE, x, end color blueE-direction.
You can start the same way, treating the start color redE, y, end color redE value as a constant. However, this time, you cannot plug in an actual constant value, like start color redE, y, equals, 2, end color redE. Instead, pretend that start color redE, y, end color redE is ​constant and take the derivative:
Or rather, since to emphasize that this is a multivariable function, we use the symbol \partial instead of d:
xf(x,y)=x(x2y3)=2xy3 \dfrac{\partial}{\blueE{\partial x}}f(\blueE{x}, y) = \dfrac{\partial}{\blueE{\partial x}}(\blueE{x}^2y^3) = 2\blueE{x}y^3
As a sanity check, you can plug in left parenthesis, start color blueE, 3, end color blueE, comma, start color redE, 2, end color redE, right parenthesis to see that we get the same result as above.
"So, what's the difference between start fraction, d, divided by, d, x, end fraction and x\dfrac{\partial}{\partial x}? They seem to be used the same way."
Honestly, as far as I'm concerned, there's not really a difference between these operations. You could be pedantic and say one is only defined for single variable functions. But as far as intuition and computation go, they are one and the same, and the difference is just meant to clarify what type of function is being differentiated.

Interpreting partial derivatives with graphs.

Consider this function:
f, left parenthesis, x, comma, y, right parenthesis, equals, start fraction, 1, divided by, 5, end fraction, left parenthesis, x, start superscript, 2, end superscript, minus, 2, x, y, right parenthesis, plus, 3,
Here is a video showing its graph rotating, just to get a feel for the three-dimensional nature of it.
Think about the partial derivative of f with respect to start color blueE, x, end color blueE, perhaps evaluated at the point left parenthesis, 2, comma, 0, right parenthesis.
fx(2,0) \dfrac{\partial f}{\blueE{\partial x}}(2, 0)
In terms of the graph, what does the value of this expression tell us about the behavior of the function f at the point left parenthesis, 2, comma, 0, right parenthesis?

Treat y as constant right arrow slice graph with plane

The first step when computing this value is to treat y as a constant. Specifically, if we are limiting our view to what happens at the point left parenthesis, 2, comma, 0, right parenthesis, we should only look at the set of points where y, equals, 0. In three-dimensional space, this set is plane perpendicular to the y-axis, passing through the origin.
Intersecting y=0 plane with the graph
This plane y, equals, 0, shown in white, slices into the graph of f, left parenthesis, x, comma, y, right parenthesis along a parabolic curve, shown faintly in red. We can interpret fx\dfrac{\partial f}{\blueE{\partial x}} as giving the slope of a tangent line to this curve. Why? Because x\partial x is a slight nudge in the x-direction, the run, and f\partial f is the resulting change in the z-direction, the rise.
What about fy\dfrac{\partial f}{\redE{\partial y}} at that same point left parenthesis, 2, comma, 0, right parenthesis? The points where start color blueE, x, equals, 2, end color blueE also make up a plane, but this time it's a plane perpendicular to the x-axis intersecting the point x, equals, 2. This slices the graph along a new curve, and fy\dfrac{\partial f}{\redE{\partial y}} will give the slope of that new curve.
Intersecting x=2 plane with the graph.
Reflection Question: In the picture to the right, the "curve" where the graph of f, left parenthesis, x, comma, y, right parenthesis, equals, start fraction, 1, divided by, 5, end fraction, left parenthesis, x, start superscript, 2, end superscript, minus, 2, x, y, right parenthesis, plus, 3 intersects the plane defined by x, equals, 2 looks like it might be a straight line. Is it really a line?
Choose 1 answer:
Choose 1 answer:

It is!
Intersecting the graph of f with the plane x, equals, 2 corresponds with treating x as the constant 2. That is, we look at f, left parenthesis, 2, comma, y, right parenthesis as a single variable function of y:
f(2,y)=15(222(2)y)+3=4545y+3=45y+195\begin{aligned} f(2, y) &= \frac{1}{5}(2^2 - 2(2)y) + 3 \\ &= \frac{4}{5} - \frac{4}{5}y + 3 \\ &= -\frac{4}{5}y + \frac{19}{5} \end{aligned}
This function is linear, so its graph is a line with slope minus, 4, slash, 5.
The partial derivative fy\dfrac{\partial f}{\partial y} evaluated at any point left parenthesis, 2, comma, y, right parenthesis gives the slope of this line, minus, start fraction, 4, divided by, 5, end fraction, no matter what y is. In fact, the partial derivative of this function doesn't depend on y at all:
fy=y(15(x22xy))=15(02x)=25x\begin{aligned} \dfrac{\partial f}{\partial y} &= \dfrac{\partial}{\partial y} \left( \frac{1}{5}(x^2 - 2xy) \right) \\ &= \frac{1}{5}(0 - 2x) \\ &= -\frac{2}{5}{x} \end{aligned}
Graphically, this means as we choose different values of x, and the plane representing the constant x value shifts left or right, it will always intersect the graph along a line, since its slope is constant with respect to y. Moreover, the slope of that line will always be minus, start fraction, 2, divided by, 5, end fraction times the value of x.

Phrasing and notation

Here are some of the phrases you might hear in reference to this fx\dfrac{\partial f}{\partial x} operation:
  • "The partial derivative of f with respect to x"
  • "Del f, del x"
  • "Partial f, partial x"
  • "The partial derivative (of f) in the x-direction"

Alternate notation

In the same way that people sometimes prefer to write f, prime instead of start fraction, d, f, divided by, d, x, end fraction, we have the following notation:
fxfxfyfyfSome variable fThat same variable\begin{aligned} f_\blueE{x} &\leftrightarrow \dfrac{\partial f}{\blueE{\partial x}} \\ f_\redE{y} &\leftrightarrow \dfrac{\partial f}{\redE{\partial y}} \\ f_{\greenE{\langle\text{Some variable }\rangle}} &\leftrightarrow \dfrac{\partial f}{\greenE{\partial \langle\text{That same variable} \rangle}} \end{aligned}

A more formal definition

Although thinking of d, x or x\partial x as really tiny changes in the value of x is a useful intuition, it is healthy to occasionally step back and remember that defining things precisely requires introducing limits. After all, what specific small value would x\partial x be? One one hundredth? One one millionth? 10, start superscript, minus, 10, start superscript, 10, end superscript, end superscript?
The point of calculus is that we don't use any one tiny number, but instead consider all possible values and analyze what tends to happen as they approach a limiting value. The single variable derivative, for example, is defined like this:
dfdx(x0)=limh0f(x0+h)f(x0)h\begin{aligned} \dfrac{{df}}{\blueE{dx}}(x_0) = \lim_{\blueE{h}\to 0} \frac{{f(x_0\blueE{+h}) - f(x_0)}}{\blueE{h}} \end{aligned}
  • h represents the "tiny value" that we intuitively think of as d, x.
  • The h0h \to 0 under the limit indicates that we care about very small values of h, those approaching 0.
  • f, left parenthesis, x, start subscript, 0, end subscript, plus, h, right parenthesis, minus, f, left parenthesis, x, start subscript, 0, end subscript, right parenthesis is the change in the output that results from adding h to the input, which is what we think of as d, f.
Formally defining the partial derivative looks almost identical. If is a function with multiple inputs, here's how that looks:
Similarly, here's how the partial derivative with respect to y looks:
The point is that h, which represents a tiny tweak to the input, is added to different input variables depending on which partial derivative we are taking.
People will often refer to this as the limit definition of a partial derivative.
Reflection question: How can we think about this limit definition in the context of the graphical interpretation above? What is h? What does it look like for h0h \to 0?

Summary

  • For a multivariable function, like f, left parenthesis, x, comma, y, right parenthesis, equals, x, start superscript, 2, end superscript, y, computing partial derivatives looks something like this:
  • This swirly-d symbol \partial , called "del", is used to distinguish partial derivatives from ordinary single-variable derivatives.
  • The reason for a new type of derivative is that when the input of a function is made up of multiple variables, we want to see how the function changes as we let just one of those variables change while holding all the others constant.
  • With respect to three-dimensional graphs, you can picture the partial derivative fx\dfrac{\partial f}{\partial x} by slicing the graph of f with a plane representing a constant y-value, and measuring the slope of the resulting cut.
Intersecting y=0 plane with the graph