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# Introduction to partial derivatives

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, squared, y, computing partial derivatives looks something like this:
\begin{aligned} \dfrac{\partial f}{\blueE{\partial x}} &= \!\!\!\!\! \underbrace{ \dfrac{\partial}{\blueE{\partial x}} \blueE{x^2}y }_{\substack{ \text{Treat }y\text{ as constant;}\\\\ \text{take derivative.} }}\!\!\!\!\! = 2\blueE{x}y \\\\ \dfrac{\partial f}{\redE{\partial y}} &= \!\!\!\!\! \underbrace{ \dfrac{\partial}{\redE{\partial y}} x^2\redE{y} }_{\substack{ \text{Treat }x\text{ as constant;}\\\\ \text{take derivative.} }}\!\!\!\!\! = x^2\cdot \redE{1} \end{aligned}
• This swirly-d symbol, \partial, often 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 start fraction, \partial, f, divided by, \partial, x, end fraction 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 $\\dfrac{df}{dx}$ 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, squared, 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 start fraction, \partial, f, divided by, \partial, x, end fraction, start fraction, \partial, f, divided by, \partial, y, end fraction, etc.
You read the symbol start fraction, \partial, f, divided by, \partial, x, end fraction out loud by saying "the partial derivative of f with respect to x".
$\begin{array}{c} \text{Can be thought of as } \\ \text{"a tiny change in} \\ \text{the function's output"} \\\\ \begin{array}{c} \begin{array}{c} \blueE{\text{Used instead of "d" in}} \\ \blueE{\text{usual }\dfrac{df}{dx}\text{ notation to}} \\ \blueE{\text{emphasize that this is}} \\ \blueE{\text{a partial derivative.}} \end{array} & \begin{array}{c} \LARGE\blueE\nearrow^{\huge\,\overbrace{\blueE\partial \greenE f}\,}\greenE \nwarrow \\\\ \Huge \text{\textemdash} \\\\ \LARGE\blueE\searrow_{\huge\,\underbrace{\blueE\partial \maroonE x}\,}\maroonE \swarrow \end{array} & \begin{array}{c} \\\\ \greenE{\text{Multivariable}} \\ \greenE{\text{function}} \\\\ \maroonE{\text{Indicates which}} \\ \maroonE{\text{input variable}} \\ \maroonE{\text{is changed slightly.}} \end{array} \end{array} \\\\ \text{Can be thought of as} \\ \text{"a tiny change in }x\text{"} \end{array}$

## Example: Computing a partial derivative

Consider this function:
f, left parenthesis, start color #0c7f99, x, end color #0c7f99, comma, start color #bc2612, y, end color #bc2612, right parenthesis, equals, start color #0c7f99, x, end color #0c7f99, squared, start color #bc2612, y, end color #bc2612, cubed
Suppose I asked you to evaluate start fraction, \partial, f, divided by, start color #0c7f99, \partial, x, end color #0c7f99, end fraction, the partial derivative with respect to x, at the input left parenthesis, start color #0c7f99, 3, end color #0c7f99, comma, start color #bc2612, 2, end color #bc2612, 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 #0c7f99, 3, end color #0c7f99, comma, start color #bc2612, 2, end color #bc2612, right parenthesis to left parenthesis, start color #0c7f99, 3, point, 01, end color #0c7f99, comma, start color #bc2612, 2, end color #bc2612, right parenthesis.
Since we only care about movement in the start color #0c7f99, x, end color #0c7f99-direction, we might as well treat the start color #bc2612, y, end color #bc2612-value as a constant. In fact, we can just plug in start color #bc2612, y, equals, 2, end color #bc2612 ahead of time before computing any derivatives:
f, left parenthesis, start color #0c7f99, x, end color #0c7f99, comma, start color #bc2612, 2, end color #bc2612, right parenthesis, equals, start color #0c7f99, x, end color #0c7f99, squared, left parenthesis, start color #bc2612, 2, end color #bc2612, right parenthesis, cubed, equals, 8, start color #0c7f99, x, end color #0c7f99, squared
Now, asking how f changes in response to a small shift in start color #0c7f99, x, end color #0c7f99 is just an ordinary, single-variable derivative.
Concept check
What is the derivative of this function f, left parenthesis, start color #0c7f99, x, end color #0c7f99, comma, start color #bc2612, 2, end color #bc2612, right parenthesis, equals, 8, start color #0c7f99, x, end color #0c7f99, squared evaluated at start color #0c7f99, x, equals, 3, end color #0c7f99?

### Without pre-evaluating $y$y

Now suppose I asked you to find start fraction, \partial, f, divided by, start color #0c7f99, \partial, x, end color #0c7f99, end fraction, 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 #0c7f99, x, end color #0c7f99, comma, start color #bc2612, y, end color #bc2612, 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 #0c7f99, x, end color #0c7f99-direction.
You can start the same way, treating the start color #bc2612, y, end color #bc2612 value as a constant. However, this time, you cannot plug in an actual constant value, like start color #bc2612, y, equals, 2, end color #bc2612. Instead, pretend that start color #bc2612, y, end color #bc2612 is ​constant and take the derivative:
start fraction, d, divided by, start color #0c7f99, d, x, end color #0c7f99, end fraction, f, left parenthesis, start color #0c7f99, x, end color #0c7f99, comma, y, right parenthesis, equals, start underbrace, start fraction, d, divided by, start color #0c7f99, d, x, end color #0c7f99, end fraction, left parenthesis, start color #0c7f99, x, end color #0c7f99, squared, y, cubed, right parenthesis, end underbrace, start subscript, start text, P, r, e, t, e, n, d, space, end text, y, start text, space, i, s, space, c, o, n, s, t, a, n, t, end text, end subscript, equals, 2, start color #0c7f99, x, end color #0c7f99, y, cubed
Or rather, since to emphasize that this is a multivariable function, we use the symbol \partial instead of d:
start fraction, \partial, divided by, start color #0c7f99, \partial, x, end color #0c7f99, end fraction, f, left parenthesis, start color #0c7f99, x, end color #0c7f99, comma, y, right parenthesis, equals, start fraction, \partial, divided by, start color #0c7f99, \partial, x, end color #0c7f99, end fraction, left parenthesis, start color #0c7f99, x, end color #0c7f99, squared, y, cubed, right parenthesis, equals, 2, start color #0c7f99, x, end color #0c7f99, y, cubed
As a sanity check, you can plug in left parenthesis, start color #0c7f99, 3, end color #0c7f99, comma, start color #bc2612, 2, end color #bc2612, 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 start fraction, \partial, divided by, \partial, x, end fraction? 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, squared, 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 #0c7f99, x, end color #0c7f99, perhaps evaluated at the point left parenthesis, 2, comma, 0, right parenthesis.
start fraction, \partial, f, divided by, start color #0c7f99, \partial, x, end color #0c7f99, end fraction, left parenthesis, 2, comma, 0, right parenthesis
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 $y$y as constant $\rightarrow$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 start fraction, \partial, f, divided by, start color #0c7f99, \partial, x, end color #0c7f99, end fraction as giving the slope of a tangent line to this curve. Why? Because \partial, x is a slight nudge in the x-direction, the run, and \partial, f is the resulting change in the z-direction, the rise.
What about start fraction, \partial, f, divided by, start color #bc2612, \partial, y, end color #bc2612, end fraction at that same point left parenthesis, 2, comma, 0, right parenthesis? The points where start color #0c7f99, x, equals, 2, end color #0c7f99 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 start fraction, \partial, f, divided by, start color #bc2612, \partial, y, end color #bc2612, end fraction 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, squared, 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?

## Phrasing and notation

Here are some of the phrases you might hear in reference to this start fraction, \partial, f, divided by, \partial, x, end fraction 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:
\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}

While it's common to refer to the partial symbol \partial as "del", this can be confusing because "del" is also the name of the Nabla symbol del, which we will introduce in the next article.

## A more formal definition

Although thinking of d, x or \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 \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:
\begin{aligned} \dfrac{{df}}{\blueE{dx}}(x_0) = \lim_{\blueE{h}\to 0} \dfrac{{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 h, \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 f, left parenthesis, x, comma, y, comma, dots, right parenthesis is a function with multiple inputs, here's how that looks:
\begin{aligned} \dfrac{\partial f}{\blueE{\partial x}}(x_0, y_0, \dots) &= \lim_{\blueE{h} \to 0} \dfrac{f(\blueE{x_0\blueE{+h}}, y_0, \dots) - f(x_0, y_0, \dots)} {\blueE{h}} \end{aligned}
Similarly, here's how the partial derivative with respect to y looks:
\begin{aligned} \dfrac{\partial f}{\redD{\partial y}}(x_0, y_0, \dots) &= \lim_{\redD{h} \to 0} \dfrac{f(x_0, \redD{y_0+h}, \dots) - f(x_0, y_0, \dots)}{\redD{h}} \end{aligned}
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 h, \to, 0?

## Summary

• For a multivariable function, like f, left parenthesis, x, comma, y, right parenthesis, equals, x, squared, y, computing partial derivatives looks something like this:
\begin{aligned} \dfrac{\partial f}{\blueE{\partial x}} &= \!\!\!\!\! \underbrace{ \dfrac{\partial}{\blueE{\partial x}} \blueE{x^2}y }_{\substack{ \text{Treat }y\text{ as constant;}\\\\ \text{take derivative.} }}\!\!\!\!\! = 2\blueE{x}y \\\\ \dfrac{\partial f}{\redE{\partial y}} &= \!\!\!\!\! \underbrace{ \dfrac{\partial}{\redE{\partial y}} x^2\redE{y} }_{\substack{ \text{Treat }x\text{ as constant;}\\\\ \text{take derivative.} }}\!\!\!\!\! = x^2\cdot \redE{1} \end{aligned}
• This swirly-d symbol \partial, often 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 start fraction, \partial, f, divided by, \partial, x, end fraction 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

## Want to join the conversation?

• Please, please can you label the axes three dimensional graphs completely and
more clearly? Your site is the best! I am 75 and trying to revise my university
math of some 50 years ago!
Many thanks
• The up axis is always z, and you can use the right hand rule to see which the other two are; your index finger will be pointing at the x-axis, and you're middle finger will be pointing at the y-axis. (Note: different software use different conventions for labeling the axes. Sometimes z represents depth and y represents height. But mac grapher - the software Grant is using here - always uses the convention I described above.)
• In Example 2, isn't cos(0*pi) = cos(0) = 1? And therefore the answer is non-negative?
• Where it says: the set of all points where y = 0 includes all points of the form (x,0,z). I believe this set is mistakenly called the xy-plane. It should be called the xz-plane.
• Hi,
I find these articles very useful, and I'd like to keep them at hand for future references. Are they available for download, in pdf or some other print-friendly format, somewhere? Thanks.
• Why there is no a mission in multivariable funtions?
(1 vote)
• It's more about learning the techniques, notation, and order of operations than plugging and cranking a bunch of examples. It is also an area of math that isn't really intended to learn purely online. This section is more of a nudge to help learn or better understand something you are currently learning in a 5th semester math course. Usually proceeded by MA 161,162 and MA261,262 or something similar. I'm just using Purdues course #s.
• I still see the gradient vary on the video graphical representation. Does not make sense.
(1 vote)
• The explanation with the graph is hard to follow. I see no relation between the graph being shown and the equations being listed. I don't understand any of it.
(1 vote)
• If you think back to Calculus 1 (or single-variable calculus), recall the the derivative of a function is equal to its slope at any point. If you don't understand that concept, it might be good to look back and review the section on derivatives. In this case, when we take a slice of the graph, the two-dimensional intersection of the graph and the plane looks like a single-variable function. Which variable the function is of depends on the orientation of the graph. If the graph is parallel to the x-axis, it looks like a function of x, and if the graph is parallel to the y-axis, the intersection looks like a function of y. The partial derivative is a way to find the slope in either the x or y direction, at the point indicated. By treating the other variable like a constant, the situation seems to simplify to something we can understand in terms of single-variable derivatives, which we learned in Calc 1. If you still do not understand, let me know, and we can try to work it out. Make sure you have learned previous lessons well.
• They kicked off with the playlist on multivariable calculus 4 or 5 years ago. Not sure about this current article though.
(1 vote)
• 2 quick questions:
1) example 2, sin( 0 + pi) = 0... however I ran my calculator and got the answer .0548. I checked and my settings are in degrees, not radians.

2) In both my homework (Dynamic Atmosphere) and your examples, when deriving 'y' for example, why do we make '2x' or 'x^2' equal to zero. Why do we not bring them as a constant?
(1 vote)
• If they are stand alone such as x^2+2x+2xy^3. If you are taking the partial derivative with respect to y, you treat the others as a constant. The derivative of a constant is 0, so it becomes
0+0+2x(3y^2). You'll notice since the last one is multiplied by Y, you treat it as a constant multiplied by the derivative of the function. I.E d/dx 2x^2 is just 2(d/dx(x^2)) or 2(2x) or 4x. Hope this helps.