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in this video I want to help you gain an

Â intuitive understanding of calculus and

Â the derivatives now maybe you're

Â thinking that you haven't seen calculus

Â since your college days and depending on

Â when you graduate maybe that was quite

Â some time back now if that's what you're

Â thinking don't worry you don't need a

Â deep understanding of calculus in order

Â to apply new networks and deep learning

Â very effectively so if you're watching

Â this video or some of the later videos

Â be wondering wow this stuff really for

Â me this calculus looks really

Â complicated my advice to you is the

Â following which is that watch the videos

Â and then if you could do the homework

Â and complete the programming homework

Â successfully then you can apply deep

Â learning in fact what you see later is

Â that in week 4 will define a couple of

Â types of functions that will enable you

Â to encapsulate everything that needs to

Â be done with respect to calculus that

Â these functions call forward functions

Â and backward functions that you learn

Â about the less you put everything you

Â need to know about counselors into these

Â functions so that you don't need to

Â worry about them anymore beyond that but

Â I thought that in this foray into deep

Â learning that this week we should open

Â up the box and peer a little bit further

Â into the details of calculus but really

Â all you need is an intuitive

Â understanding of this in order to build

Â and successfully apply these algorithms

Â oh and finally if you are among that

Â maybe smaller group of people that are

Â expert in calculus if you're very

Â familiar with calculus observe this it's

Â probably okay for you to skip this video

Â but for everyone else let's dive in and

Â try to get an intuitive understanding of

Â derivatives I've plotted here the

Â function f of a equals 3/8 so it's just

Â a straight line to gain intuition about

Â derivatives let's look at a few points

Â on this function let's say that a is

Â equal to 2 in that case f of a which is

Â equal to 3 times 8 is equal to 6 so if a

Â is equal to 2 then you know F of a will

Â be equal to 6 let's say we give the

Â value of a you know just a little bit of

Â a nudge I'm going to just bump up me a

Â little bit so there is now 2.00 1 right

Â so I'm going to get a like a tiny little

Â nudge to the right so now is let's say 2

Â oh one this plug this is to scale 2.01

Â the 0.001 difference is too small to

Â show on this plot this give them a

Â little nudge to the right now f of a is

Â equal to three times at so six point

Â zero zero three Simplot this over here

Â this is not the scale this is six point

Â zero zero three so if you look at this

Â low triangle here some highlighting in

Â green what we see is that if I match a

Â 0.001 to the right then F of a goes up

Â by 0.03 the amount that F of a went up

Â is three times as big as the amount that

Â I judged a to the right so we're going

Â to say that the slope of the derivative

Â of the function f of a at a equals two

Â or when a is equal to 2 the slope this

Â reading and you know the term derivative

Â basically means slope is just that

Â derivative sound like a scary a more

Â intimidating word whereas slope is a

Â friendlier way to describe the concept

Â of derivative so one of these year

Â derivative just think slope of the

Â function and more formally the slope is

Â defined as the height divided by the

Â width of this little triangle that we

Â have in green so this is you know 0.03

Â over 0.01 and the fact that the slope is

Â equal to 3 or the derivative is equal 3

Â just represents the fact that when you

Â watch a to the right by 0.01 by tiny

Â amount the amount that F of a goes up is

Â three times as big as the amount that

Â United the inertial a in the horizontal

Â direction so that's all that the slope

Â of a line is now let's look at this

Â function at a different point let's say

Â that a is now equal to five

Â in that case f of a three times a is

Â equal to 15 so let's say I again give a

Â and notch to the right

Â a tiny longnecks is now bumped up to

Â five point over one F of a is three

Â times that

Â so f of a is equal to fifteen point zero

Â three and so once again when I bump into

Â the right not a to the right by 0.001

Â F of a goes up three times as much

Â so the slope again at a equals five is

Â also three so the way we write is that

Â the slope of the function f is equal to

Â three we say D F of a da and this just

Â means the slope of the function f of a

Â when you nudge the variable a a tiny

Â little amount um this is equal to three

Â and an alternative way to write this

Â derivative formula is as follows you can

Â also write this as d da of f of a so

Â whether you put the f of a on top of

Â whether you write it you know down here

Â it doesn't matter

Â but all those equation means is that if

Â I nudge a to the right a little bit

Â I expect F of a to go up by three times

Â as much as I not just the value of

Â little a now for this video I explained

Â derivatives talking about what happens

Â we nudge the variable a by 0.001 um if

Â you want the formal mathematical

Â definition of the derivatives

Â derivatives are defined with an even

Â smaller value of how much energy a to

Â the right so it's not open over 1 is not

Â 0.001 is not 0.0 and so on 1 is sort of

Â even smaller than that and the formal

Â definition of derivative says what have

Â you nudge a to the right by an info

Â testable amount basically an infinite

Â infinitely tiny tiny amount if you do

Â that does f of a go up three times as

Â much as whatever was a tiny tiny tiny

Â amount that you now stay to the right so

Â that's actually the formal definition of

Â a derivative but for the purposes of our

Â intuitive understanding we're going to

Â talk about nudging a to the right by

Â this small amount 0.001 even if it's

Â 0.001 isn't exactly you know tiny tiny

Â insa testable now one property of the

Â derivative is that no matter where you

Â take the slope of this function it is

Â equal to 3 whether a is equal to 2 or a

Â is equal to 5 the slope of this function

Â is equal to 3 meaning that whatever is

Â the value of a if you increase it by

Â 0.001 that

Â value of f of a goes up by three times

Â as much so this function has the same

Â slope everywhere and one way to see that

Â is that wherever you draw this your

Â little triangle right the height divided

Â by the width always has a ratio of three

Â to one so I hope this gives you a sense

Â of what the slope what the derivative of

Â the function means for a straight line

Â where in this example the slope of the

Â function was three everywhere in the

Â next video let's take a look at a

Â slightly more complex example where the

Â slopes of the function can be different

Â at different points on the function

Â