Lecture notes · Calculus II · Lecture 11

Chain Rule

MIT 18.02 Multivariable Calculus, lecture 11, explained in plain English

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Part 1 · 0:00–10:00

This part introduces differentials, a way of writing small changes in a function with several variables and using them to estimate how the function changes.

Key ideas

The lecture starts from implicit differentiation in single-variable calculus, which relates small changes in y to small changes in x. For example, starting from x = sin(y), you can find the derivative of inverse sine. 1:26
The total differential adds up every cause that can change a function of several variables. For f(x, y, z), it is df = f_x dx + f_y dy + f_z dz, where each term is a partial derivative times a small change. 4:20
A differential like df is not a number. The change delta f is a number, and df is only a placeholder that you fill with other things. 5:51
If you put small numbers in for delta x, delta y, and delta z, the right side gives an estimate of delta f. This estimate is the linear, or tangent plane, approximation. It is close to the true change but not exactly equal to it. 7:19
If x, y, and z all depend on a parameter t, you can divide everything by dt. This gives the rate at which f changes as t changes. 9:15

Words to know

differential: A small change written with a d in front of a variable, such as dx or df. It is a symbol you can manipulate, not a plain number.
total differential: The full sum of all the small changes that can move a function, with one term for each input variable, each multiplied by how much that input affects the output.
partial derivative: How fast a function changes when you change only one input and hold the others still. It is written like f_x or the partial f over partial x.
implicit differentiation: A method for finding a derivative when y is tied to x through an equation, not written directly as y = a formula. You take the change of both sides and solve for the rate you want.

Example

Think of the total cost of a trip as depending on the distance you drive and the price of gas. If the price changes a little and the distance changes a little, the total change in cost is roughly the change from each part added together. This is the same idea as the total differential, which adds the effect of each input.

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

Part 2 · 10:00–20:00

This part explains why the chain rule works, using small changes and the idea that a tiny step in one thing causes a tiny step in another.

Key ideas

If f depends on x, y, and z, and each of those depends on time t, then the rate of change of f with respect to t is found by adding up each part: how f changes with x times how fast x changes, and so on. 10:43
This rule is called the chain rule. It tells you how to find the rate of change of a value that depends on something that itself depends on something else. 11:09
The lecturer warns that the textbook mixes up d and delta in a couple of formulas. Those two symbols should not be mixed. 12:19
The first proof uses differentials, which the lecturer says we should not fully trust yet. A better way uses approximation formulas, which we already trust. Change time by a small amount delta t, and see how x, y, and z change. 16:32
Divide everything by delta t, then let delta t get smaller and smaller. The ratios of changes approach the derivatives, so the approximation becomes exact in the limit. 18:19

Words to know

differential: A tiny change in a value, written with a d, such as dx or df. The lecturer says we should be careful trusting these until the proof is done.
infinitesimal: Extremely small, so small it is almost zero but not quite. It is used to describe very small changes.
derivative: How fast something changes when another thing changes, such as how fast position changes with time.
limit: The value a ratio gets closer and closer to as the change we look at gets smaller and smaller.

Example

Imagine a shop's profit depends on how many items it sells, and the number sold depends on the day of the week. To find how fast profit changes from day to day, you combine how profit changes with sales and how sales change with the day. This is the same idea the chain rule uses.

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

Part 3 · 20:00–30:00

This part shows how the chain rule works on a worked example, then uses it to explain where the product rule and quotient rule come from.

Key ideas

The chain rule helps when a value depends on other values that each depend on t. You find how the outer value changes with each piece, then multiply by how that piece changes with t, and add the results. 21:06
For w = x^2 y z with x = t, y = e^t, and z = sin t, the chain rule gives 2t e^t + t^2 e^t + cos t. Plugging the values in first and taking the ordinary derivative gives the same answer. 23:00
The chain rule is worth using when the formula is very complicated, or when you only know the partial derivatives and not the full formula. 23:20
A function of two variables, u and v, can be written as the product uv. Taking the chain rule step by step, with u and v each depending on t, gives back the usual product rule. 27:21
The same method works for the quotient rule. The lecture starts this by writing g = u divided by v and finding the partial derivative of g with respect to u, which is one over v. 29:16

Words to know

chain rule: A method for finding how a value changes when it depends on other values that themselves change. You work through each link in the chain, one step at a time.
partial derivative: How a formula changes when you change only one of its inputs and hold the others still. The lecture writes it as a curly d, like partial w over partial x.
product rule: The rule for finding the derivative of two things multiplied together. The derivative of uv is u times the derivative of v, plus v times the derivative of u.
quotient rule: The rule for finding the derivative of one thing divided by another. The lecture treats it as a practice problem after the product rule.

Example

Think of a bakery where the number of loaves sold depends on how many people walk past, and that depends on the hour of the day. If you want to know how sales change over time, you multiply the change in loaves per person by the change in people per hour, and that chain of changes gives you the answer.

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

Part 4 · 30:00–40:00

This part shows how to find how a quantity w changes when the variables it depends on, x and y, themselves depend on two other variables, u and v.

Key ideas

The lesson gives a real case: a function written in polar coordinates (r and theta) that you want to convert to ordinary x and y coordinates, or the other way around. You want to know how the derivatives relate without doing messy substitutions. 32:27
Plugging the formulas straight in can be messy, especially when arctangent appears. So it is often easier to work with derivatives instead. 33:10
Start with the differential dw = f_x dx + f_y dy. Then replace dx and dy with their own differentials, written in terms of du and dv. This gives dw in terms of du and dv only. 35:25
The number in front of du is the partial derivative of w with respect to u, meaning the rate of change with u while v is held fixed. The number in front of dv works the same way for v. 37:28
To remember the chain rule, follow the dependencies: w depends on x and y, and x and y depend on u. So each term multiplies a partial of w with respect to x or y by a partial of x or y with respect to u. 39:22

Words to know

partial derivative: The rate at which a quantity changes when you change one input, while you keep all the other inputs fixed.
differential: A small change in a quantity, written as a small step like dx or dw, that shows how much the output moves for tiny changes in the inputs.
polar coordinates: A way to locate a point using a distance r from the center and an angle theta, instead of x and y positions.
arctangent: The inverse of the tangent function. It gives an angle back from a ratio, and it can make formulas look complicated.

Example

Think of a shop's profit, which depends on its sales and its costs. Sales themselves depend on the price and the amount spent on ads. To see how profit reacts to a change in ad spending, you follow the chain: ads change sales, and sales change profit. The chain rule adds up those effects along each path.

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

Part 5 · 40:00–50:07

This part explains the chain rule for functions of several variables, and why partial derivatives must be handled carefully.

Key ideas

If a function depends on x and y, and x and y depend on u, you can find how the function changes with u by multiplying each partial rate through x and y and adding the results. 40:00
Each multiplication is one chain of cause and effect: a small change in u changes x, which changes f, and the product gives that effect. The same is true through y. 41:00
When u changes, x and y both change at once, so the total effect on f is the sum of the two effects. 41:43
You cannot cancel partial symbols the way you cancel ordinary d's, because partial derivatives measure change with one variable at a time. 42:25
The gradient vector is a way to bundle all the partial derivatives of a function into one vector. 49:23

Words to know

partial derivative: How fast a function changes when you change one input and hold all the other inputs still. It is written with a curly d.
total derivative: How fast a function changes when all of its inputs change together. It is written with a straight d.
chain rule: A rule for finding the rate of change of something that depends on another thing, which in turn depends on a third thing. You multiply the rates along the way and add up the paths.
gradient vector: A list of numbers, one for each input, made from the partial derivatives of a function. It packs them together into one object.

Example

Imagine your monthly budget depends on rent and food costs, and both of those depend on how many people live in your house. If one more person moves in, rent and food both change at the same time. The total effect on your budget is the sum of those two effects, just as the lesson adds the effects through x and y.

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

← Lecture 8: Partial Derivatives · Lecture 12: Gradient →

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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