The Chain Rule

Summary

In this StatQuest video, Josh Starmer explains the chain rule, a fundamental concept in calculus. Starting with a review of derivatives using basic examples, he builds up to more complex scenarios involving exponential and square root functions. Starmer uses relatable analogies, like predicting shoe size based on weight and height, to demonstrate how the chain rule connects different relationships. He concludes with an example from machine learning, showing how the chain rule helps minimize the squared residuals in a loss function. The clear, step-by-step approach simplifies the concept for viewers.

Takeaways

• 📚 The video explains the chain rule, assuming the viewer is familiar with derivatives.
• 📉 A parabola is used to explain how the derivative gives the slope of the tangent, showing how 'awesomeness' changes with respect to liking StatQuest.
• 🧮 The chain rule is illustrated using a simple example of predicting height from weight, and shoe size from height.
• 🔗 The chain rule connects two relationships: height based on weight and shoe size based on height.
• 📐 The derivative of shoe size with respect to weight is found by multiplying the derivative of height with respect to weight and the derivative of shoe size with respect to height.
• 🚶‍♂️ A more complex example involving hunger and craving for ice cream is presented, showing how to use the chain rule to find how craving changes over time.
• 🔄 The video emphasizes the application of the chain rule even when equations are not in a simple, separate form, using parentheses to clarify relationships.
• 📊 A practical example of applying the chain rule in machine learning is given, focusing on residual sums of squares to find the best fit line for weight and height data.
• 🧩 The chain rule is repeatedly applied by separating equations into simpler components to compute derivatives efficiently.
• 🎯 The video concludes with finding the intercept that minimizes squared residuals to determine the best fit line, demonstrating how the chain rule helps in optimizing functions.

Q & A

What is the chain rule in calculus?

The chain rule is a fundamental concept in calculus that allows us to compute the derivative of a composite function by multiplying the derivatives of the inner and outer functions.

Why is the chain rule important in the context of the examples provided?

The chain rule helps connect changes between multiple variables, as demonstrated in the examples with weight, height, and shoe size, as well as hunger and craving for ice cream. It allows us to understand how changes in one variable affect another through an intermediary variable.

How does the chain rule apply to the weight, height, and shoe size example?

In the example, the chain rule shows how weight indirectly affects shoe size through height. The derivative of shoe size with respect to weight is calculated as the product of the derivative of shoe size with respect to height and the derivative of height with respect to weight.

What is the relationship between the slope and the derivative in the provided examples?

The slope of a line represents the rate of change between two variables, and this is the same as the derivative in the examples. For instance, the slope of the green line between weight and height is 2, so the derivative of height with respect to weight is also 2.

How does the chain rule simplify complex derivative calculations?

The chain rule breaks down complex composite functions into simpler parts by differentiating the outer function first and then multiplying it by the derivative of the inner function. This is useful when dealing with nested functions, such as in the ice cream craving and hunger example.

Why is the chain rule especially useful in the example with hunger and craving for ice cream?

The chain rule simplifies the process of calculating how ice cream cravings change with respect to time since the last snack by considering how hunger changes with time and how cravings change with hunger. Without the chain rule, the calculation would be more complex and less intuitive.

How does the chain rule help in machine learning applications, like calculating the residual sum of squares?

In machine learning, the chain rule helps compute the derivative of the loss function, such as the residual sum of squares, by breaking down the derivative into simpler parts, making it easier to find the optimal parameters (e.g., the intercept) that minimize the loss.

What is the significance of using parentheses in the chain rule examples?

Parentheses help isolate the inner function or 'stuff inside' in a composite function, making it easier to apply the chain rule by clearly identifying the inner and outer functions for differentiation.

What role does the power rule play in the chain rule examples?

The power rule is used in combination with the chain rule to differentiate functions that involve powers, such as the square of a variable. It simplifies finding the derivative of a function raised to a power, which is a common occurrence in the examples.

How does the video explain the process of minimizing the squared residual in machine learning?

The video explains that minimizing the squared residual involves finding the derivative of the squared residual with respect to the intercept and setting it to zero. The chain rule is used to calculate the derivative by considering the relationship between the residual and the intercept.

Outlines

00:00

🎓 Introduction to the Chain Rule

The video begins with Josh Starmer introducing the topic of the chain rule in calculus. He assumes the viewer has a basic understanding of derivatives and aims to provide a deeper explanation of the chain rule. Using simple examples, such as how a parabolic curve represents the relationship between 'likes StatQuest' and 'awesomeness,' Josh reviews the concept of derivatives, covering the power rule for determining the slope of a tangent line at any point along the curve. This segment serves as a foundation for understanding the chain rule, which is the main focus of the video.

05:02

📏 Understanding the Chain Rule through Height, Weight, and Shoe Size

The second section introduces a practical example to explain the chain rule. Josh uses weight, height, and shoe size data to show how changes in one variable affect another through intermediate steps. He highlights how changing weight predicts height, which in turn predicts shoe size. The chain rule is introduced by explaining how the derivative of shoe size with respect to weight is calculated through the product of two derivatives: height with respect to weight and shoe size with respect to height. This process simplifies complex relationships and provides a clear understanding of how the chain rule works.

10:03

🍦 Chain Rule in Action: Craving Ice Cream Based on Hunger and Time

In this example, Josh demonstrates how the chain rule is applied in situations where relationships are more complex, such as hunger and craving ice cream. As time since the last snack increases, hunger and ice cream cravings change at different rates, and Josh fits exponential and square root functions to this data. The chain rule helps solve for the derivative of cravings with respect to time by breaking down the problem into manageable parts. Josh emphasizes how intermediate variables, like hunger, link time and cravings, simplifying what would otherwise be a difficult derivative to compute.

15:05

📊 Chain Rule for Complex Equations and the Sum of Squares

This section extends the application of the chain rule to more complex equations, such as those encountered in machine learning when minimizing loss functions like the residual sum of squares. Josh walks through an example where height and weight data are used to fit a line to measurements. He explains how adjusting the intercept affects the residual, and how the chain rule is used to find the derivative of the squared residual with respect to the intercept. By following the steps of the chain rule, Josh demonstrates how this process leads to determining the best-fitting line for the data.

Keywords

💡Chain Rule

The chain rule is a fundamental concept in calculus used to calculate the derivative of a function composed of multiple functions. In the video, it explains how the derivative of a composed function is the product of the derivatives of each function. For example, when predicting shoe size based on weight via height, the chain rule helps to link these variables.

💡Derivative

A derivative represents the rate at which a function changes at any point, essentially indicating the slope of a curve. The video demonstrates derivatives through examples such as how 'awesomeness' changes with respect to 'likes statquest,' and how the slope of a line reflects the change in variables like height and weight.

💡Slope

The slope refers to the steepness or incline of a line and is calculated as the ratio of the vertical change to the horizontal change. In the video, slope is a key part of understanding how height changes with respect to weight, and how shoe size changes with respect to height. For example, the slope of the line between weight and height is given as 2.

💡Power Rule

The power rule is a basic rule in calculus used to differentiate functions of the form x^n. In the video, this rule is used to explain the derivative of 'likes statquest squared,' which simplifies to 2 times 'likes statquest.' The power rule is also applied in more complex examples, such as differentiating the hunger and ice cream craving functions.

💡Exponential Function

An exponential function describes situations where growth or decay accelerates rapidly. In the video, the exponential function is used to model hunger levels over time, showing that people get hungrier at a faster rate as more time passes since their last snack. This leads into how the chain rule can simplify the derivative calculation for such functions.

💡Square Root Function

The square root function is used in the video to model the relationship between hunger and craving for ice cream. As hunger increases, the craving initially increases but eventually tapers off, following a square root function. This helps to illustrate how non-linear relationships between variables can be differentiated using the chain rule.

💡Residual

In statistical modeling, a residual represents the difference between an observed value and the value predicted by a model. The video explains how residuals are used to assess how well a model fits the data, and introduces the concept of squared residuals for minimizing error in predictions, particularly in the example involving fitting a line to height and weight measurements.

💡Squared Residuals

Squared residuals are the squares of the residuals, used to minimize differences between observed and predicted values in models. In the video, minimizing squared residuals is key to determining the best-fitting line for height and weight data. By squaring the residuals, larger errors are penalized more heavily, leading to a more accurate model.

💡Intercept

The intercept is the value where a line crosses the y-axis when x is zero. In the video, the intercept is manipulated in the equation for height and weight to find the best fit. By adjusting the intercept, the video shows how the residuals and squared residuals change, ultimately aiming for an intercept that minimizes the squared residuals.

💡Loss Function

A loss function quantifies the error in a predictive model, and the goal is to minimize this error. The video uses the example of the residual sum of squares, a common loss function in machine learning, to show how the chain rule helps in calculating derivatives to minimize the loss and achieve the best model fit.

Highlights

Introduction to the chain rule and a quick review of basic derivative concepts.

Using a parabola to illustrate the relationship between 'likes Statquest' and 'awesomeness,' with a review of the power rule.

Explanation of how the derivative provides the slope of the tangent line, showing the rate of change of awesomeness with respect to Statquest likes.

A basic example using weight, height, and shoe size to explain how the chain rule links different variables, allowing for predictions.

Demonstration of calculating derivatives by linking height to weight and shoe size, with a clear application of the chain rule.

Detailed breakdown of how the slope between variables (weight, height, and shoe size) helps explain their derivatives.

The essence of the chain rule: The derivative of shoe size with respect to weight is the product of two derivatives (shoe size with height, and height with weight).

A more complex example showing how hunger is related to time since the last snack and cravings for ice cream, with an exponential model and a square root function.

Explanation of how the chain rule simplifies complex derivative calculations when hunger links the time since the last snack to ice cream cravings.

Rewriting complex equations to make the chain rule more apparent, by focusing on parts of the equation that can be grouped in parentheses.

Illustration of how the chain rule applies to the residual sum of squares, a common loss function in machine learning.

Finding the derivative of the residual squared with respect to the intercept using the chain rule.

Explanation of the connection between the residual and intercept, and how the derivative of the residual squared helps minimize errors in a model.

Using the chain rule to minimize the residual sum of squares and find the best fitting line in regression analysis.

Final conclusion summarizing how the chain rule works across different examples, from simple functions to more complex machine learning applications.

Transcripts

the chain rule is cool

stat quest yeah

[Music]

hello i'm josh starmer and welcome to

statquest

today we're going to talk about the

chain rule

and it's going to be clearly explained

note this stat quest assumes that you

are already familiar with the basic

idea of a derivative and just want a

deeper understanding of

the chain rule

that said let's do a super quick review

imagine we collected these measurements

from a bunch of people

on the x-axis we measured how much they

liked statquest

and on the y-axis we measured

awesomeness

we can then fit this orange parabola to

the data

the equation for the parabola is

awesomeness

equals likes statquest squared

the derivative of this equation tells us

the slope of the tangent line at any

point along the curve

the slope of the tangent line tells us

how quickly

awesomeness is changing with respect to

like's stat quest

we can calculate the derivative of

awesomeness with respect to like's

stat quest by using the power rule

the power rule tells us to multiply

like's stat quest

by the power which is 2 and raise stat

quest by the power

2 -1 and since

minus one equals one and raising

something by one

is the same as omitting the power we end

up with

two times like's statquest

okay bam that's the review

now let's dive into the

chain rule

with a super simple example

imagine we collected weight and height

measurements from three people

and then we fit a line to the data

now if someone tells us they weigh this

much

we can use the green line to predict

that they are this

tall bam now imagine we collected height

and shoe size measurements and we fit a

line to the data

now if someone tells us that they are

this tall

we can use the orange line to predict

that this

is their shoe size bam

now if someone tells us that they weigh

this much

then we can predict their height and we

can use the predicted height

to predict shoe size and if we change

the value for weight

we see a change in shoe size

bam

now let's focus on this green line that

represents the relationship between

weight

and height we see that for every one

unit increase in weight

there's a two unit increase in height

in other words the slope of the line is

2 divided by 1 which equals 2

and since the slope is 2 the derivative

the change in height with respect to a

change in weight

is two now since the slope of the green

line

is the same as its derivative two

the equation for height is height

equals the derivative of height with

respect to weight

times weight which equals two

times weight note

the equation for height has no intercept

because the green line goes through the

origin

now let's focus on the orange line that

represents the relationship between

height and shoe size in this case

we see that for every one unit increase

in height

there is a one-quarter unit increase in

shoe size

and i admit that it's hard to see the

one-quarter unit increase in shoe size

so just trust me anyway

because we go up one quarter unit for

every one unit we go

over the slope is one quarter

divided by one which equals one quarter

and since the slope is one quarter the

derivative

or the change in shoe size with respect

to a change in

height is one quarter

now since the slope of the orange line

is the same as its derivative

the equation for shoe size is

shoe size equals the derivative of shoe

size with respect to height

times height which equals one-quarter

times height and again

because the orange line goes through the

origin the equation for shoe size has no

intercept now because

weight can predict height

and height can predict shoe size

we can plug the equation for height into

the equation for shoe size

now if we want to determine exactly how

shoe size

changes with respect to changes in

weight

we can take the derivative of shoe size

with respect to weight and the

derivative

of the equation for shoe size with

respect to weight

is just the product of the two

derivatives

in other words because height connects

weight

to shoe size the derivative of shoe size

with respect to weight

is the derivative of shoe size with

respect to height

times the derivative of height with

respect to weight

this relationship is the essence of the

chain rule

plugging in numbers gives us one half

and that means for every one unit

increase in weight

beep boop beep there is a one-half

unit increase in shoe size bam

now let's look at a slightly more

complicated example

imagine we measured how hungry a bunch

of people were

and how long it had been since they last

had a snack

as time since the last snack increases

on the x-axis

people got hungrier and hungrier at a

faster rate

so we fit an exponential line with

intercept one-half

to the measurements to reflect the

increasing rate of hunger

then we measured how much people craved

ice cream and how hungry they

were the hungrier someone was

the more they craved ice cream

but after a certain amount of hunger the

craving did not continue to increase

very much

so we fit a square root function to the

data to reflect how the increase in

craving

tapers off now if we want to see how the

rate of

craving ice cream changes with respect

to the time

since the last snack plugging the

equation for hunger

into the equation for craves ice cream

gives us an equation without an obvious

derivative

to convince yourself that taking the

derivative of this

is no fun at all pause the video and

give it a try

however because hunger links time since

last snack

to craves ice cream we can use

the chain rule to solve for this

derivative

first the power rule tells us that the

derivative of hunger

with respect to the time since the last

snack is

two times time

likewise the power rule tells us that

the derivative of craves ice cream with

respect to hunger is

one divided by two times the square root

of hunger

so with these two derivatives

the chain rule tells us that the

derivative of craves ice cream

with respect to time is

the derivative of craves ice cream with

respect to hunger

times the derivative of hunger with

respect to time since last snack

so we plug in the derivatives

and plug in the equation for hunger

and cancel out the twos

and we get the derivative of craves ice

cream with respect to time

since last snack this derivative

tells us how quickly or slowly our

craving for ice cream

changes with respect to time

double bam

in this last example it was obvious that

hunger was the link between time since

last snack and craves ice cream

and we had an equation for hunger in

terms of time

and an equation for craves ice cream in

terms of hunger

however usually these relationships are

not so obvious

instead of having two separate equations

we usually get the first equation jammed

into the second

and when all you have is this it's not

so

obvious how the chain rule applies

so we can talk about how to apply the

chain rule

in this situation let us scooch the

equation to the left so we have room to

work

now one thing we can do in this

situation is look for things in the

equation that can be put

in parentheses for example

the square root symbol can be replaced

with parentheses

now we can say that the stuff inside the

parentheses

is time squared plus

one half and craves ice cream

can be rewritten as the square root of

the stuff inside

now the chain rule tells us that the

derivative of craves ice cream

with respect to time is

the derivative of craves ice cream with

respect to the stuff

inside times the derivative of the stuff

inside

with respect to time the power rule

gives us the derivative of craves ice

cream with respect to the stuff

inside and the power rule gives us the

derivative of the stuff inside

with respect to time now we just plug

the derivatives

into the chain rule and plug in the

equation for the stuff inside

cancel out the twos and we get the

derivative of craves ice cream

with respect to the time since last

snack

and that's exactly what we got before

bam

now let's look at how the chain rule

applies to the residual sum of squares

a commonly used loss function in machine

learning

note if this does not make any sense to

you

just imagine i said now let's look at

one last example

imagine we measured someone's weight and

height

and we wanted to fit this green line to

the measurement

now to keep things simple let's assume

we can only move the green line

up and down the equation for the green

line

is height equals the intercept

plus 1 times weight and we can change

the intercept

but to keep things simple we can't

change the slope

which is set to 1. if we set the

intercept to 0

then this location on the green line is

the predicted height

and we can calculate the residual the

difference between the observed height

and the value predicted by the line

and we can plot the residual on this

graph

which has the intercept on the x-axis

and the residual on the y-axis

if we change the intercept here

then we can see the change in the

residual here

and because a common way to evaluate how

good the green line fits the data

is the squared residual we can plot the

squared residual

here where we have the residuals on the

x-axis

and the squared residuals on the y-axis

now if we change the intercept here

then we change the residual here and

here

and changing the residual here changes

the squared residual

here in order to find the value for the

intercept that minimizes the squared

residual

we are going to find the derivative of

the squared residual

with respect to the intercept and then

we're going to find where the derivative

equals zero because given the function

y equals the residual squared the

derivative

is zero at the lowest point

the chain rule says that because the

residual links the intercept to the

squared residual

then the derivative of the squared

residual with respect to the intercept

is the derivative of the squared

residual with respect to the residual

times the derivative of the residual

with respect to the intercept

the power rule tells us that the

derivative of the residual squared

is just two times the residual

so let's plug that in to solve for the

derivative of the residual

with respect to the intercept we move

the equation for the residual

over here so we have room to work

then we plug in the equation for the

predicted height

then in order to remove these

parentheses

we multiply everything inside by

negative one

now the derivative of the residual with

respect to the intercept

is zero because this term does not

contain the intercept

plus negative one because the derivative

of the negative

intercept equals negative one plus zero

because the last term does not contain

the intercept

now do the math and we are left with

negative one

and that makes sense because the

derivative is just the slope of the

orange line

and by i we can see that the slope of

the orange line

is negative one so let's plug this

derivative

in here and do a little math

and plug in the equation for the

residual

now we have the derivative for the

residual squared in terms of the

intercept

note if instead of starting with

separate equations for the residual

and the residual squared we started with

just the equation for the residual

squared with the equation for the

predicted height

jammed into it then just like before

we can use parentheses to help us out

in this case we'll call everything

between the outermost parentheses

the stuff inside which equals the

observed

minus the intercept minus one times

weight

and that means the residual squared can

be rewritten

as the square of the stuff inside

now we can use the chain rule to

determine the derivative of the residual

squared

with respect to the intercept it's the

derivative of the residual squared with

respect to the stuff inside

times the derivative of the stuff inside

with respect to the intercept

just like before the derivative of the

residual

with respect to the stuff inside is two

times the stuff inside so we plug that

into the chain rule

and the derivative of the stuff inside

with respect to the intercept

is negative one so we plug that into the

chain rule now we just plug in the stuff

inside

multiply two with negative one

and we end up with the exact same

derivative as before

bam now we want to find the value for

the intercept

such that the derivative of the residual

squared equals zero

so we plug in the observed height and

the observed weight

set the derivative equal to 0

and solve for the intercept

and at long last we see that when the

intercept

equals one we minimize the squared

residual

and we have the best fitting line

triple bam hooray

we've made it to the end of another

exciting stack quest

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until next time quest on