Chain Rule

MIT 18.01 Single Variable Calculus, lecture 4, explained in plain English

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Part 1 · 0:00–10:00

This part introduces the product rule, a formula for finding the derivative of two functions multiplied together, and shows why it works.

Key ideas

The product rule says the derivative of u times v is u'v + uv'. You take the derivative of each piece in turn, so a product becomes a sum. 2:43
The professor applies the rule to x^n times sin x. He takes the derivative of x^n, keeps sin x as it is, then keeps x^n and takes the derivative of sin x. 3:06
To prove the rule, look at how much the product changes when x changes by a small amount, called delta x. Then rearrange the terms so the changes in u and v appear. 4:20
Divide the change by delta x, then let delta x shrink toward 0. This limit step gives the derivative. 7:42
The step where v(x + delta x) becomes v(x) works because v is continuous, meaning it has no sudden jumps. 9:09

Words to know

derivative: A measure of how fast a function's output changes as its input changes.
difference quotient: The change in a function divided by the change in x. It is the starting point for finding a derivative.
limit: The value something gets closer and closer to as a small amount shrinks toward zero.
continuity: A function is continuous when its graph has no sudden jumps or breaks, so small changes in input give small changes in output.

Example

Think of u and v as two numbers that are multiplied together. If one of them changes a little, the product changes by an amount that depends on both the change and the other number. The product rule describes exactly how those two effects add up.

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Part 2 · 10:00–20:00

This part finishes the product rule and teaches the quotient rule, which tells you how to find the derivative of one function divided by another.

Key ideas

The product rule comes from a change in uv. Most terms cancel, and dividing what is left by delta x and taking the limit gives the derivative. 10:00
The quotient rule says the derivative of u/v is (u'v - uv') divided by v squared. The lecturer calls it a strange formula but shows where it comes from. 11:50
When delta x shrinks to 0, delta v also shrinks to 0, because v is continuous. That is why the limit gives a clean formula. 16:04
A special case is 1 divided by v, where u is the constant 1. Its derivative is minus v' divided by v squared. 17:36
Using v = x^n, the derivative of 1 divided by x^n is -n times x to the power of (-n - 1). This is the rule for negative powers of x. 18:51

Words to know

quotient: The result of dividing one number or function by another.
reciprocal: One divided by something. For example, the reciprocal of v is 1/v.
continuous: A function with no sudden jumps or breaks, so small changes in x make only small changes in the output.
limit: The value a result gets closer and closer to as a small number, like delta x, gets closer and closer to zero.

Example

Think of a bike that goes 60 miles and takes 2 hours, so its speed is 60 divided by 2. If you want to know how fast that speed changes when the time changes a little, you need the quotient rule. The top and bottom each change a little, and the rule tells you how those two changes combine.

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Part 3 · 20:00–30:00

This part shows how to find the derivative of a function built from one function inside another, using the chain rule.

Key ideas

The power rule you already know also works for negative exponents. The lecturer says it comes from the quotient rule, so it applies to negative powers as well as positive ones. 21:31
A composition is a function made by doing one step and then another. You can handle it by giving the inside step a new name, such as x, so the problem becomes two simpler steps. 23:33
The chain rule says the derivative of a composition is a product: dy/dt = (dy/dx) times (dx/dt). You can think of the dx terms as canceling, though that is a memory aid rather than a full proof. 25:42
To use the rule on y = (sin t)^10, first differentiate the outside function, which gives 10x^9. Then differentiate the inside function, which gives cos t. 27:19
Because you introduced x yourself, you must put back x = sin t at the end. The final answer is 10 sin^9(t) times cos t. 28:11

Words to know

composition: Putting one function inside another, so the output of the first becomes the input of the second.
derivative: A measure of how fast a value changes when another value changes a little.
intermediate variable: An extra letter, like x here, that stands for the middle step between the start and the end of a problem.
chain rule: The rule for finding the derivative of a composition: multiply the derivative of the outside function by the derivative of the inside function.

Example

Think of making toast from bread. First you slice the bread, then you toast the slices. To describe the whole job, you name the middle result, the sliced bread, and track how each step changes it. The chain rule works the same way: it multiplies the change from each step together.

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

Part 4 · 30:00–40:00

This part explains how to take the derivative of a function inside another function (the chain rule), how to take higher derivatives, and what the derivative notation means.

Key ideas

A composite function is one function placed inside another, such as sin(10t). The chain rule handles these by splitting the problem into an outside function and an inside function. 30:16
To differentiate sin(10t), first take the derivative of the outside function (sine becomes cosine) and keep the inside unchanged. Then multiply by the derivative of the inside function, which is 10, to get 10cos(10t). 31:49
A higher derivative just means differentiating again and again. The derivative of a derivative is the second derivative, and the next one is the third derivative. 32:57
For u = sin x, differentiating four times gives back sin x. Each step moves sine to cosine, then to minus sine, then to minus cosine, and then back to sine. 35:47
The symbol d/dx can be read as an operator, meaning an instruction you apply to a function to get a new function. Capital D is another name for it, and writing d/dx twice means taking the second derivative. 37:06

Words to know

composite: A function built by putting one function inside another, like sin(10t), where 10t is inside the sine.
derivative: A new function that shows how fast the original function is changing at each point.
operator: A symbol or instruction you apply to a function to get a new function, like d/dx meaning 'differentiate this'.
notation: The set of symbols people use to write a math idea, such as u', du/dx, or D.

Example

Think of a chain of two gears: a motor turns gear A, and gear A turns gear B. How fast gear B spins depends on both links, so you combine the two speeds by multiplying them. The chain rule works the same way for a function placed inside another function.

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

Part 5 · 40:00–45:59

This part shows how to write derivatives in different ways, and how repeated derivatives of x^n work out to a fixed number, n factorial.

Key ideas

The notation d squared u over dx squared is not d of x squared. It means the second derivative, where dx is squared as a whole, and it is one of several ways to write the same idea. 40:00
The professor says the different notations all mean the same thing. The third derivative can be written as d cubed u over dx cubed, or as the operator D done three times on u. 40:37
Taking the fourth derivative gives back the original function only for sines and cosines. This is not true for functions in general. 41:20
When you take the derivative of x^n again and again, the power drops by one each time and the exponent numbers multiply together. After n steps you get a constant, n times n-1 and so on down to 1, which is called n factorial. 42:06
The derivative of n factorial is 0, because the derivative of any constant is 0. So the next derivative after n steps is zero. 45:37

Words to know

derivative: A measure of how fast a function is changing. Taking a derivative is the act of finding that rate of change.
operator: A symbol that tells you to do an action to a function, such as taking a derivative. The capital D here is an operator.
constant: A fixed number that does not change, such as 5 or n factorial. Its derivative is always 0.
mathematical induction: A proof method where you show a pattern works for the first case, then show that if it works for one case it works for the next. The professor says the x^n calculation was an example of it.

Example

Think of a ladder where each step down takes you one level lower, and you count the steps you took. Each derivative of x^n is like one step down, lowering the exponent by 1. After n steps you reach a flat, fixed value, and one more step leaves you at zero, just as a flat floor has no slope.

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

← Lecture 3: Derivatives · Lecture 6: Exponential and Log →

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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