Subtitles section
Repeat sentence
Subtitles mode
Full screen
Previous
Next
x 1.0
Playback speed
More

Product rule for more than two functions

Video vocabulary

result

Copy vocabulary

US / rɪ'zʌlt /

UK / rɪˈzʌlt /

A2
  • noun
  • Something produced through tests or experiments
  • Final score of a competition, election or game
  • verb
  • To be the outcome of other causes and effects

everything

Copy vocabulary

US / 'evrɪθɪŋ /

UK / ˈɛvriˌθɪŋ /

A1
  • pronoun
  • All of the things mentioned

they

Copy vocabulary

US / ðeɪ /

UK / ðe /

A1
  • pronoun
  • Two or more people, animals, or things

way

Copy vocabulary

US / weɪ /

UK / we /

A1
  • adverb
  • Used to emphasize the amount of something
  • noun
  • Particular aspect of something being thought about
  • One of various choices that could be made
  • Particular existing condition or state
  • (Often long) distance or length of time
  • How something is done
  • Route to go to a place
  • Small street
  • Usual behavior, emotions or habits of someone

close

Copy vocabulary

US / kləʊz /

UK / kloʊz /

A1
  • adjecitve
  • Almost; near
  • (Of a friend) well-known; liked by others
  • Being near or related
  • Similar; nearly the same
  • verb
  • To come progressively nearer to something
  • To end; to bring to an end
  • To make something shut; stop from being open

take

Copy vocabulary

US / teɪk /

UK / tek /

A1
  • noun
  • Filming of a single scene for TV or the movies
  • verb
  • To accept something that is offered
  • To use a method of traveling, e.g. a bus or train
  • To use a camera to make (a photo)
  • To pick up something and go away with it
  • To study a subject in school
  • To attempt to pass (an exam)
  • To require a certain amount of time, money, space

expression

Copy vocabulary

US / ɪk'spreʃn /

UK / ɪkˈsprɛʃən /

A2
  • noun
  • Act of making your thoughts and feelings known
  • Group of words that have a specific meaning

pretty

Copy vocabulary

US / 'prɪtɪ /

UK / ˈprɪti /

A1
  • adjecitve
  • Being attractive to the eye in a simple way
  • adverb
  • Very; to a great degree

bit

Copy vocabulary

US / bɪt /

UK / bɪt /

A1
  • noun
  • Device put in a horse's mouth to control it
  • Small piece of something
  • verb
  • (E.g. of fish) to take bait and be caught

derivative

Copy vocabulary

US / dɪˈrɪvətɪv /

UK / dɪˈrɪvətɪv /

B1
  • adjecitve
  • Not from original ideas; from somewhere else

multiply

Copy vocabulary

US / ˈmʌltɪplaɪ /

UK / ˈmʌltəˌplaɪ /

B1
  • verb
  • To increase in number by reproducing/having babies
  • To increase something by a large quantity
  • To have increased by a large quantity
  • To calculate two numbers e.g. 2 x 4

rewrite

Copy vocabulary

US / ˌri:'raɪt /

UK / riˈraɪt /

A2
  • verb
  • To write again to improve, or correct errors

all

Copy vocabulary

US / ɔ:l /

UK / ɔl /

A1
  • adverb
  • Completely; totally
  • determiner
  • Being every one of something
  • pronoun
  • 100% of something; the complete amount

just

Copy vocabulary

US / dʒʌst /

UK / dʒʌst /

A1
  • adjecitve
  • Proper or appropriate; as is deserved
  • Doing or being what is right or fair
  • Legally correct; having enough evidence
  • adverb
  • Almost; nearly
  • Exactly at the time of
  • Exactly (the same as, like)

time

Copy vocabulary

US / taɪm /

UK / taɪm /

A1
  • noun
  • Speed at which music is played; tempo
  • Point as shown on a clock, e.g. 3 p.m
  • Number of hours, minutes needed to do something
  • Occasion when something happens
  • Period or occasion that something occurred
  • Period in history or the past
  • Something measured in minutes, hours, days, etc.
  • How long an event takes; duration
  • verb
  • To check speed at which music is performed
  • To choose a specific moment to do something
  • To measure how long an event takes, e.g. a race
  • To schedule something to occur at a specific moment

standard

Copy vocabulary

US / 'stændəd /

UK / ˈstændəd /

A2
  • adjecitve
  • Being the accepted normal level of quality
  • (Of a language) being the most accepted in a place
  • noun
  • Official unit of measuring something
  • Principle of behaving in a moral way

stuff

Copy vocabulary

US / stʌf /

UK / stʌf /

A1
  • noun
  • Generic description for things, materials, objects
  • verb
  • To push material inside something, with force

well

Copy vocabulary

US / wel /

UK / wɛl /

A1
  • adjecitve
  • Being in a fortunate, good condition or state
  • Healthy and in good condition
  • adverb
  • In an acceptable, proper, or successful manner
  • In a complete, total, or full manner
  • A great and considerable degree
  • In a successful way, in a good or satisfactory way
  • exclamation
  • Used to begin or continue a particular statement
  • Used to show you want to start saying something
  • Used to express that you are waiting for a reply
  • You say this to change the topic of a conversation
  • Used to indicate you are making a final remark
  • noun
  • Long deep hole in the ground to get water
  • verb
  • To cause your eyes to fill with tears

white

Copy vocabulary

US / waɪt /

UK / hwaɪt, waɪt /

A1
  • adjecitve
  • Belonging to a race of people with light skin
  • Of the color of fresh snow and milk
  • noun
  • Person's name
  • Lightest color; The color of fresh snow.

four

Copy vocabulary

US / fɔ:(r) /

UK / fɔr, for /

A1
  • number
  • 4

imagine

Copy vocabulary

US / ɪ'mædʒɪn /

UK / ɪˈmædʒɪn /

A1
  • verb
  • To think creatively about; form mental picture of

here

Copy vocabulary

US / hɪə(r) /

UK / hɪr /

A1
  • adverb
  • In this place; at this location, position or time

will

Copy vocabulary

US / wɪl /

UK / wɪl /

A1
  • noun
  • Desire or choice of someone
  • Document saying who gets your money when you die
  • Strong desire to do something; determination
  • Desire to do something; strong wish to do
  • other
  • Used with verbs to express the future
  • verb
  • To leave property to someone after your death
  • To use mental effort to make something happen
  • Am (is, are) likely to do
  • To influence someone to do something
  • To make something happen by strongly wanting it

use

Copy vocabulary

US / ju:s /

UK / ju:s /

A1
  • noun
  • Ability to employ something
  • Need to enable it to work
  • Permission to do something with something
  • Act of spending money
  • Operation of a machine for its function
  • verb
  • To have; to eat, drink, consume etc.
  • To need to operate or work
  • To operate a machine, e.g. a computer
  • To treat a person badly
  • To do something with, for a task or purpose

write

Copy vocabulary

US / raɪt /

UK / raɪt /

A1
  • verb
  • To compose letters and words on paper or a screen
  • To invent or create a computer program
  • To create a song or piece of music

plus

Copy vocabulary

US / plʌs /

UK / plʌs /

A2
  • noun
  • A benefit or advantage
  • preposition
  • In addition; add
  • In addition to; Indicates a number above
  • Having a value greater than zero

equal

Copy vocabulary

US / 'i:kwəl /

UK / ˈikwəl /

A1
  • adjecitve
  • Same in shape, size, or number
  • verb
  • To add up or be the same as

neat

Copy vocabulary

US / ni:t /

UK / nit /

A2
  • adjecitve
  • That you approve of; cool
  • (Of alcohol) not weakened with water or a mixer
  • Not messy; clean and orderly; simple and effective

now

Copy vocabulary

US / naʊ /

UK / naʊ /

A1
  • adverb
  • At the present time or moment
  • Right away or in the next possible moment
  • Used to refer to something that just happened
  • Time between the present and a moment in the past
  • conjunction
  • Resulting from or because of something

apply

Copy vocabulary

US / ə'plaɪ /

UK / əˈplaɪ /

A1
  • verb
  • To spread a substance or liquid over a surface
  • To commit your time and effort to doing something
  • To make something useful in a certain situation
  • To be relevant for a situation
  • To ask formally for (job, permission etc.)

but

Copy vocabulary

US / bət /

UK / bʌt,bət /

A1
  • conjunction
  • Used before you say something different, opposite

distribute

Copy vocabulary

US / dɪˈstrɪbju:t /

UK / dɪˈstrɪbjut /

A2
  • verb
  • To hand out or give to people
  • To send products to stores to sell
  • To spread something over an area

color

Copy vocabulary

US / 'kʌlə(r) /

UK / ˈkʌlɚ /

A1
  • noun
  • Quality of things you can see, e.g. red, blue
  • Pink or red in your face, e.g. after being ill
  • The shade of someone's skin e.g. black, or yellow
  • verb
  • To change or affect someone's opinion
  • To make something colorful using colored pencils

our

Copy vocabulary

US / 'aʊə(r) /

UK / aʊr /

A1
  • determiner
  • Belonging to us

not

Copy vocabulary

US / nɒt /

UK / nɑt /

A1
  • adverb
  • Word indicating the negative
  • (Used to form the negative of verbs)

want

Copy vocabulary

US / wɔnt /

UK / wɑnt, wɔnt /

A1
  • noun
  • Something you desire or wish for
  • State or condition of poverty
  • verb
  • To desire or wish for something; hope for a thing

video

Copy vocabulary

US / 'vɪdɪəʊ /

UK / ˈvɪdiˌo /

A2
  • adjecitve
  • Of a moving film, with sound
  • noun
  • Recording capturing action with sound
  • verb
  • To make a moving film of, with sound

would

Copy vocabulary

US / wʊd /

UK / wʊd /

A1
  • other
  • Used to indicate something imagined
  • Used to ask or request something politely
  • Verb used to indicate possibility, request etc.
  • Used to indicate a preference
  • Used to indicate a reaction
  • Indicates something that occurred repeatedly
  • Used to indicate willingness or ability to do

can

Copy vocabulary

US / kæn /

UK / kən,kæn /

A1
  • noun
  • A metal container for storing food
  • other
  • Able to; have the skill to; have the time to
  • verb
  • To have permission to
  • To be possible to do
  • To preserve food by sealing it in a metal container

essentially

Copy vocabulary

US / ɪˈsenʃəli /

UK / ɪˈsenʃəli /

A2
  • adverb
  • Basically; (said when stating the basic facts)

right

Copy vocabulary

US / raɪt /

UK / raɪt /

A1
  • adjecitve
  • Correct or true
  • The south side of you looking at the rising sun
  • adverb
  • Exactly in a place or time
  • Being appropriate for a particular event
  • In a straight or direct manner
  • noun
  • Something you are legally or morally allowed to do
  • Ideal of what is just and good
  • Political view based on conservatism
  • verb
  • To fix something previously wrong or unjust
  • To make upright, as a boat that has turned over

about

Copy vocabulary

US / ə'baʊt /

UK / əˈbaʊt /

A1
  • adverb
  • To be ready to; going to
  • Nearly; approximately; roughly
  • Near; next to; close to
  • In every direction; in the area of
  • preposition
  • Concerning or of (a topic)

again

Copy vocabulary

US / ə'ɡen /

UK / əˈɡɛn /

A1
  • adverb
  • One more time, once more

that

Copy vocabulary

US / ðæt /

UK / ðæt, ðət /

A1
  • adjecitve
  • The person or thing being talked about
  • Used to identify something both the speakers know
  • adverb
  • Used to reinforce adjectives and adverbs
  • To such a degree
  • conjunction
  • Used to connect a noun clause
  • Used to introduce an adverbial clause
  • Used to introduce a noun clause
  • determiner
  • Used as a noun to refer to something
  • pronoun
  • Referring an object far away
  • Used to refer to the relative pronoun ‘which’

what

Copy vocabulary

US / wɒt /

UK / hwɑt, hwʌt, wɔt, wʌt,hwət, wət /

A1
  • pronoun
  • Thing or things that; the same kind as
  • Used to emphasize something you are about to say
  • adjecitve
  • Used to refer to a particular amount, or number
  • Used to place emphasis on a particular quality
  • adverb
  • To a certain degree
  • determiner
  • Question words used for asking for information
  • exclamation
  • Used to express excitement, shock, or surprise
  • Used to ask people to repeat something they said

each

Copy vocabulary

US / i:tʃ /

UK / itʃ /

A1
  • determiner
  • Every one of two or more things
  • pronoun
  • Per one person or thing

respect

Copy vocabulary

US / rɪ'spekt /

UK / rɪˈspɛkt /

A2
  • noun
  • Behavior that follows the established rules
  • Admiration for someone
  • Regard or admiration for someone or something
  • verb
  • To follow the established rules
  • To think very highly of another person

then

Copy vocabulary

US / ðen /

UK / ðɛn /

A1
  • adverb
  • A time that is not now
  • Following another thing in time or arrangement
  • Following on reasonably from what was just stated
  • At that time, not now

tell

Copy vocabulary

US / tel /

UK / tɛl /

A1
  • verb
  • To (strongly) advise a person to do something
  • To be able to guess
  • To say or communicate information to someone
  • To inform, but without speaking; indicate
  • To have a negative effect on someone
  • To order someone to do something; command
  • To recognize something based on certain signs
  • To report something to an authority
  • To explain something or teach someone something

first

Copy vocabulary

US / fɜ:st /

UK / fɚst /

A1
  • adjecitve
  • 1st; Coming before all others in time or place
  • noun
  • Something that has not happened before
  • Position of the winner of a competition

rest

Copy vocabulary

US / rest /

UK / rɛst /

A1
  • noun
  • Time when one relaxes, sleeps, or is inactive
  • State in which there is a lack of motion
  • Part of something remaining after some was used
  • verb
  • To stop working or avoid activity to recover
  • (Of dead person) to lie in a grave
  • To lean, support or place something against
  • To not be tired because you had enough sleep

over

Copy vocabulary

US / 'əʊvə(r) /

UK / ˈovɚ /

A1
  • adverb
  • Finished; no longer happening
  • Towards the ground/away from a standing position
  • noun
  • A set of six consecutive balls bowled in cricket
  • other
  • Finished
  • preposition
  • Showing the relation when a thing is above another
  • On or covering something
  • Above or across something
  • More than you need
  • From one side to the other
  • In the direction of; toward

and

Copy vocabulary

US / ənd /

UK / ənd, ən,ænd /

A1
  • conjunction
  • Used to refer to two or more things
  • Plus; in addition; on top of that
  • Used to introduce an action that follows another

two

Copy vocabulary

US / tu: /

UK / tu /

A1
  • number
  • 2

where

Copy vocabulary

US / hwɛə /

UK / hwɛr, wɛr /

A1
  • conjunction
  • At or in the place which
  • In whatever place
  • pronoun
  • Word you use to ask the location of something

finally

Copy vocabulary

US / ˈfaɪnəli /

UK / ˈfaɪnəli /

A1
  • adverb
  • used especially at the beginning of a sentence to introduce the last point or idea
  • After a long time or some difficulty

think

Copy vocabulary

US / θɪŋk /

UK / θɪŋk /

A1
  • verb
  • To have an idea about something without certainty
  • To have an idea, opinion or belief about something

could

Copy vocabulary

US / kəd /

UK / kʊd /

A1
  • other
  • Past form of 'can' to mean have ability
  • Used to ask for something politely, or to offer

know

Copy vocabulary

US / nəʊ /

UK / noʊ /

A1
  • verb
  • To be familiar with a person or place, thing
  • To feel sure or be convinced about something
  • To discover or be aware of something
  • To be able to distinguish
  • To have knowledge of things

with

Copy vocabulary

US / wɪð /

UK / wɪð, wɪθ /

A1
  • preposition
  • As an employee of a certain company
  • Against something or someone
  • By using something as a means
  • Due to; because of
  • Including
  • Possessing as a feature
  • Related to
  • (Of the manner/attitude by which you do something)
  • Used to express that people or things are together

focus

Copy vocabulary

US / 'fəʊkəs /

UK / ˈfokəs /

A2
  • noun
  • Main purpose or center of interest or activity
  • verb
  • To see clearly by adjusting your eyes or a camera
  • To keep your attention, concentration on one thing

prime

Copy vocabulary

US / praɪm /

UK / praɪm /

A2
  • adjecitve
  • Concerning the highest in quality or value
  • Concerning the highest level of achievement
  • Most important, chief importance in value
  • verb
  • To get a person ready for; prepare someone
  • To coat with paint to prepare it for the main coat

have

Copy vocabulary

US / həv /

UK / hæv /

A1
  • verb
  • To drink, smoke, eat or use something
  • To experience the effects of something
  • To organize an event, or join in an activity
  • To own, possess, or hold something
  • To cause to happen or produce a particular effect

part

Copy vocabulary

US / pɑ:t /

UK / pɑ:rt /

A1
  • adjecitve
  • To some degree; half; not fully
  • noun
  • Division of a book
  • Ratio of something, e.g. 3 of gin, 1 of tonic
  • A line in a person's hair, made with a comb
  • Character an actor plays on stage, movie, or TV
  • Some, but not all of a specific thing
  • Role in causing something to happen
  • verb
  • To make a line in a person's hair, by using a comb
  • To move two things apart creating a space
  • To leave someone, or to break off a relationship

other

Copy vocabulary

US / 'ʌðə(r) /

UK / ˈʌðər /

A1
  • adjecitve
  • A thing different from that mentioned
  • determiner
  • (Something) else; not the first (one)
  • pronoun
  • Being the one and only thing of that category

one

Copy vocabulary

US / wʌn /

UK / wʌn /

A1
  • pronoun
  • You; we; people in general; someone
  • The thing or person previously referred to
  • adjecitve
  • Used to refer to people generally
  • Being a strong example of (something mentioned)
  • number
  • Number 1

how

Copy vocabulary

US / haʊ /

UK / haʊ /

A1
  • adjecitve
  • To what degree? in what amount?
  • adverb
  • (Used to explain the way to do something)
  • exclamation
  • For what purpose or reason?
  • (Used to ask someone or something's condition)

kind

Copy vocabulary

US / kaɪnd /

UK / kaɪnd /

A1
  • adjecitve
  • In a caring and helpful manner
  • friendly and considerate
  • noun
  • One type of thing

you

Copy vocabulary

US / jʊ /

UK / ju /

A1
  • pronoun
  • Person someone is speaking or writing to
  • Person or people in general

this

Copy vocabulary

US / ðɪs /

UK / ðɪs /

A1
  • adjecitve
  • Used to indicate something already been discussed
  • Used to indicate something happening around now
  • adverb
  • An indication by a physical motion or gesture
  • A specific amount of
  • determiner
  • Person, thing, or idea near you
  • pronoun
  • Person, thing that has already been discussed

separately

Copy vocabulary

US / ˈseprətli /

UK / ˈsɛpərɪtlɪ /

A2
  • adverb
  • Not with another person or thing

term

Copy vocabulary

US / tɜ:m /

UK / tɚm /

A1
  • noun
  • Conditions applying to an agreement, contract
  • Length of time something is expected to happen
  • Fixed period of weeks for learning at school
  • The (precise) name given to something
  • verb
  • To call; give a name to

three

Copy vocabulary

US / θri: /

UK / θri /

A1
  • number
  • Number 3

little

Copy vocabulary

US / 'lɪtl /

UK / ˈlɪtl /

A1
  • adjecitve
  • Being short in length of time
  • Not great in size, amount, or degree; small
  • Being very young or younger than others
  • adverb
  • By an amount or degree that is not very much
  • Not often; rarely
  • noun
  • A person's name

let

Copy vocabulary

US / let /

UK / lɛt /

A1
  • verb
  • To allow someone to do something
  • Introducing a suggestion to do something together
  • To rent a house, etc. to others

the

Copy vocabulary

US / ðə /

UK / ðə /

A1
  • article
  • Used to refer to something already mentioned
  • Used to show there is only one of something
  • determiner
  • Used to describe all of a family
  • Subtitles section
  • What I want to do in this video is think about how we can take the derivative of an expression that can be viewed as a product not of two functions but of three functions.
  • And we're going to do it using what we know of the product rule.
  • And the way we could think about it is we can view this as the product, first, of two functions, of this function here and then that function over there.
  • And then separately take the derivative.
  • So if we just view the standard product rule, it tells us that the derivative of this thing will be equal to the derivative of f of x-- let me close it with a white bracket-- times the rest of the function.
  • So times g of x-- let me close it with the-- times g of x times h of x times plus just f of x times the derivative of this thing.
  • Times the derivative with respect to x of g of x times h of x.
  • Let me write that a little bit neater.
  • But what is this thing right over here going to be equal?
  • Well we can apply the product rule again.
  • So here, I'm just focusing on this part right over here.
  • The derivative of this is just going to be g prime of x times h of x plus g of x times the derivative of h, times h prime of x.
  • So everything that we had the derivative of g of x times h of x is this stuff right over here.
  • And so we're going to multiply that times f of x.
  • So let's rewrite all of this stuff here.
  • So this first term right over here we can rewrite.
  • So all of this is going to be equal to f prime of x-- that's that right over there-- times g of x times h of x plus-- And now we're going to distribute this f of x.
  • So it's f of x times this plus f of x times this.
  • So f of x times this is f of x times g prime of x, the derivative of g, g prime of x, times h of x.
  • Let me do that in that white color.
  • And then finally, f of x times this is just f of x times g of x times h prime of x.
  • And this was a pretty neat result.
  • Essentially, we can view this as the product rule where we have three, where we could have our expression viewed as a product of three functions.
  • Now we have three terms.
  • In each of these terms, we take a derivative of one of the functions and not the other two.
  • Here we took the derivative of f.
  • Here we took the derivative of g.
  • Here we took the derivative of h.
  • And you can imagine, if you had the product of four functions here, you would have four terms.
  • In each of them you'd be taking a derivative of one of the functions.
  • If you had n functions here, then you would have n terms here.
  • And in each of them you would take the derivative of one of the functions.
  • So this is kind of a neat result.