Partial Derivatives

MIT 18.02 Multivariable Calculus, lecture 8, explained in plain English

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Part 1 · 0:00–10:00

This part introduces functions of two or more variables and explains how to picture them as surfaces in space.

Key ideas

A function of one variable depends on one input, like f(x) = sin(x), and its graph is a curve. A function of two variables depends on two inputs, x and y, and gives one output number, written f(x, y). 1:08
Some functions only work for certain inputs. The set of inputs where a function gives a valid answer is called its domain of definition. For example, the square root of y only works when y is zero or positive. 2:44
To draw a function of two variables, you plot a point at height z = f(x, y) above each (x, y) spot. All those points together form a surface in space. 6:31
Real-world functions can depend on many variables. Temperature on Earth depends on position, elevation, and time, so it needs more than two inputs. Drawing more than two inputs is hard, but the same tools still work. 4:12
The lecture uses z = -y as a first example. Its graph is a flat plane. Changing x does nothing, because x does not appear in the formula. 8:39

Words to know

function of two variables: a rule that takes two input numbers, x and y, and gives back one output number.
surface: a flat or curved sheet in space, like the top of a hill or a sheet of paper bent into a shape.
domain of definition: the set of inputs where a formula gives a valid answer, such as avoiding dividing by zero.
graph: a picture that shows all the input and output pairs of a function, drawn in space.

Example

Think of the temperature at places on Earth. You pick a spot using longitude and latitude, which are two numbers, and the temperature there is the output. Plot that temperature above each spot and you get a bumpy surface, like a map with hills of heat and cold.

3 checked practice questions for this part are in the app.

Part 2 · 10:00–20:00

This part shows how to picture a 3D surface by slicing it, and how a contour plot works as a flat map of the same surface.

Key ideas

To picture a surface like z = 1 - x^2 - y^2, you can work piece by piece. Cut it with flat slices and see what shape each slice makes. 10:00
Setting x = 0 gives z = 1 - y^2, which is a parabola pointing downward. Setting z = 0 gives the circle x^2 + y^2 = 1, which has radius 1. 12:16
A contour plot is a flat map of a surface. Each curve on it marks the points where the function has the same value, like a height on a real map. 15:47
A level curve comes from slicing the graph with a flat horizontal plane at one height. Squashing many level curves together gives the contour plot. 18:31
Each line on a contour plot is a path you can walk along without going up or down. Closer lines mean the surface changes height faster. 19:38

Words to know

parabola: A smooth U-shaped curve. It opens upward or downward, like the path of a thrown ball.
saddle: A surface that curves up in one direction and down in the other, like a horse saddle or a mountain pass.
level curve: A line on a flat map that connects all the points where a function has the same value.
contour plot: A flat picture of a 3D surface, drawn as many level curves, like a geographic map of hills.

Example

Think of a hiking map of a mountain. Each line on the map joins places at the same height, so you can walk along one without going up or down. The contour plot of a function works the same way, with each line showing one fixed value.

3 checked practice questions for this part are in the app.

Part 3 · 20:00–30:00

This part teaches how to read contour plots, which are maps of level curves that show where a function has the same value, and how spacing and shape reveal steepness and hills, valleys, and passes.

Key ideas

A contour plot shows level curves. Each curve joins all the points where the function has the same value, like a topographic map joins points at the same height. 20:21
To find the level curve for a number, set the function equal to that number and solve. For f(x,y) = -y, the level 1 curve is y = -1, a horizontal line. 24:26
When the curves are evenly spaced, the function changes at a steady rate. When they are close together, the function changes fast, which means the ground is steep. 24:47
For 1 - x² - y², the level curves are circles. They get closer together as you move out, so the surface starts flat at the top and gets steeper. 27:52
For y² - x², the zero level curve is two crossing diagonal lines. This shape matches a saddle point, like a mountain pass, where you can stay at the same height in four directions. 29:18

Words to know

level curve: A line on a map where every point has the same value. On a hiking map, it is a line of constant height.
contour plot: A map made of level curves, or colored bands, that shows how a function's value changes across a space.
saddle point: A point shaped like a mountain pass. Walking one way it goes up, and walking another way it goes down.
surface of revolution: A 3D shape made by spinning a curve around a center line, like a bowl or a dome. Its level curves are circles.

Example

Think of a topographic hiking map. Lines that are close together mean a steep hill, and lines that are far apart mean flat land. If you walk along one line, you stay at the same height, just as a level curve keeps the function's value the same.

3 checked practice questions for this part are in the app.

Part 4 · 30:00–40:00

This part shows how a contour map can tell us whether a function goes up or down as we move, and introduces partial derivatives, which measure change along one variable at a time.

Key ideas

A contour plot is a map of a function. Moving east-west or north-south on the map is the same as changing x and y, so the map shows how the function's value changes as you move. 31:49
From a contour plot you can quickly see whether the function goes up or down in each direction. This is a rough, qualitative check, not an exact amount. 33:17
To measure how fast the function changes, you need a derivative. For a function of one variable, the derivative is the slope of the tangent line on its graph. 36:19
The derivative gives a good guess for nearby values: the value at x0 plus the derivative times the change in x. 37:24
With two variables, the change can depend on whether you move x or y. So we use partial derivatives, written with a curly d, which change only one variable at a time while the others are held fixed. 38:36

Words to know

partial derivative: A rate of change measured while only one input changes and the other inputs stay fixed.
contour plot: A flat map where curved lines join points that have the same function value, like height lines on a hiking map.
derivative: How fast a function's value changes when its input changes a tiny bit.
tangent line: A straight line that just touches a curve at one point and has the same steepness as the curve there.

Example

Imagine a hill and you are standing on it. If you walk east, the ground may go up steeply, but if you walk north, it may stay almost flat. A partial derivative tells you how steep the hill is in just one direction, like walking only east or only north.

3 checked practice questions for this part are in the app.

Part 5 · 40:00–46:13

This part explains partial derivatives, which measure how a function changes when only one of its variables changes at a time.

Key ideas

A function with several variables has no single usual derivative. Instead, it has one partial derivative for each variable. 40:00
To find the partial derivative with respect to x, you change only x a tiny bit and keep y the same. Then you look at the rate of change. 40:43
Geometrically, you fix y, slice the graph with a vertical plane, and take the slope of that slice. That slope is the partial derivative with respect to x. 42:30
To compute a partial derivative in practice, treat the other variables as constants and use the usual derivative rules on the one you care about. 44:17

Words to know

partial derivative: The rate of change of a function when you change only one variable and hold all the others fixed.
differentiable: A function is differentiable when its partial derivatives exist. Most functions the lecturer will use are like this.
slice: Cutting through a 3D graph with a flat plane, which leaves a curve you can study.
tangent line: A straight line that just touches a curve at one point and has the same slope as the curve there.

Example

Imagine you are standing on a hill and you walk straight east. How steeply the ground rises or falls depends only on your east-west step, not on your north-south position. This is an everyday analogy for holding one variable fixed while changing the other (the lesson does not use this example).

3 checked practice questions for this part are in the app.

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Notes written by AI (Claude) from the lecture's official transcript, so check anything important against the lecture. MIT OpenCourseWare · CC BY-NC-SA 4.0 · not affiliated with or endorsed by MIT. These notes are shared under the same license.

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