let's do some math for fun this is about derivative in calculus and let me ask you
guys this if I tell you the derivative of F times G is the derivative F times theta of G how would
you feel you guys for laugh at me right because you guys know this is incorrect however this is
not always incorrect sometimes it may be true huh interesting right so the question is we are going
to find functions F and G so that this is indeed correct how come to that okay now let's see what
we want this to happen and you know that when we take interruptive of a product there's the
legitimate product rule right why don't we write that down first and the person down to write down
is this I will keep the first function right here and then we will differentiate the second
function so f times G prime and we added with we keep the second function and we multiply by
Y derivative of the first which is f prime this is the determine product rule and we'll have to
set this to be the fake pottery so a set is equal to F prime times G prime okay and now what can we
do we are forcing this right here the original F the original G F prime G prime right so why
don't we try to isolate something out of this and since idea for example on the left hand side now
I don't think I can do anything at all right but then here we see we have F times G prime and then
on the right hand side we also have the G prime if I move this to the left hand side we can fight all
the cheap right right you can do the same thing you can keep this and then move this tree there
and so on but let me just move this to the left hand side so I can factor the G Prime this is
symmetrical anyway so how do you put your what you want to do it so I'm going to move this left
hand side I'll keep this right here first okay so we'll have F times G prime and the - this
is f prime times G prime and I'll bring this to the right hand side so it will be a minus G times
Prime like this okay and as I said I will factor out the G prime and usually when i factor things
are like this I like to put it at the end because the prime notation sometimes I get study confusing
so right here I already done F minus F prime first and then G prime like this okay and this is equal
to negative G x prime and now why don't we collect all the g's together G a G PI and also the F and F
prime together so let me divide both side by G so that this end our cancels or D PI D right here as
well and how would be PI goes up by this F minus F prime so divided by F - prime on the reference
as well so that is that what ends up and then we will see the left hand side is G prime over G and
this is equal to negative F Prime on the top over F minus F Prime right alright so we are making
some progress and you see that I'm going to just change the little bit right here because after
negative here I know if I switch the words of the subtraction on the bottom I can look at this this
negative parenthesis and now put down F prime first and a minus F can switch the order of the
subtraction and negate it so that can cancel the negative right so I can do caduceus the positive
version s have prime over that F prime minus F so I don't worry about the negative that's
good so now we have this is equal to that and not what do we recognize the neck inside the function
in the denominator and interruptive is on the top now it's the derivative of a log function
right but we eat the natural log what we can do is we can integrate okay so let's do that and
assuming dependent variable is X as usual so when you do it right here on the left hand side you
will get natural log of G of X okay once again if you differentiate natural log you get 1 over
the inside and then times that the routine set so you get this and usually we put a plus C but then
don't worry about it because I will keep this as how this we start the integration on the right
hand side so radius as the integral F prime over F prime minus F DX so now how can I get a G function
this is Ln of G so let me just do a to this power and then eat with that so that it cancel so
finally let me write out G and this time going to be more legitimate by using X for the independent
variable you ready dance G of X this is equal to e raised to the power and the power integral I don't
know what is but I already does the integral of f prime of x over F prime of X minus F bags yes
this is the connection between F and G okay that's what we can do so what does this mean well let me
show you that any more space so so now let me show you guys how we can use this formula to get F and
G so that if fake prada rule actually works as you can see G of X is actually depending on what do we
choose for f of X and we just have to work out this integral and e to the power okay this is a
pretty crazy integral right so be sure to choose something not too crazy I'm going to show you an
example not too easy not too hard just about right let me just say I'm going to select so I will say
that my function f of X to be x squared okay and you can try to work with other functions but let
me do this for you guys when we have f of X is equal to x squared we know F prime of X will be 2x
okay and now I just have to plug in this and that into this formula and work out the G so we know
G of X will be e 2 PI over the powers we have the integral F prime is 2 x over F prime is another 2x
and the minus f of X is x squared so this is what we have okay and we just have to work that out so
that's C this is going to be e and then the power is the integral that's crazy huh to work this out
as you can see I can factor out an X right so let me factor the X in the denominator and we
have 2 minus X like this so that they cancel and this is e and the integral we will just have the
two let me bring your 2 to the front and we have 1 over 2 minus X DX like that right this is going
to be e and then the 2 is doing the front of the integral so what's this integral of 1 over 2 minus
X it's going to be Ln 2 minus X right button be sure you know that the relative of 2 minus X is
negative 1 divided by negative 1 so you have a negative right here you've done that with that
okay so to the use substitution if you need to and this is for trendabl can we simplify more
yeah but we carefully though bring the number here to the power here so this is going to look like e
to the Ln and then this is 2 minus X and you have this raised to the negative 2 power like that and
finally we can cancel the end the Ln we couldn't do it earlier here ensure you have e and now it
has to be a nice number just one okay so that you know this is just 2 minus X and then raised
to the negative 2 power which in other word is 1 over 2 minus X and raised to the past due to power
and this is G of X ok then let me tell you this is going to blow my check this out if you so earlier
I was telling you guys if you want to take the derivative of F times G and the fake prada rule
is to just the ripe AB and then multiply by the derivative G now let's do it carefully if you want
to differentiate well f is x squared so let me put that down times G is that right so we can write
that down as 1 over 2 minus X square like this in fact this is going to be really interesting this
is going to be F prime is going to be 1 that's 2 X so we know that and then we just need to multiply
by the derivative G prime well let's get at the root of G prime ok so you should write this as 2
minus X and then squared I shall have to keep that early on like that huh negative 2 by the
way so to get G prime you know you bring the power to the front so you have negative 2 times 2 minus
X and then subtract once we have negative 3 but don't poke your x the exact derivative which is
negative 1 so all all negative 2 times negative 1 is partly 2 and this is 2 minus x raised to the
negative 3 power if you would like you can write this as 2 on the top over parenthesis 2 minus
X raised to the past is 3 power if you put this down for G prime which is then ready down 2 over
2 minus X raised to the third power let me tell you this is correct of course you can read this
down nicely as 2 times 2 which is 4 and then you have the X on the top over that right 2 minus X to
the third power this is the cool way to cool this way to do the relative don't believe me now let's
do the normal way if you were like so you know to differentiate this a normal way what you will
have to do is you keep the first function which is x squared and you differentiate the second
function multiplied right directive this which is staff so you just multiply which is 2 over that 2
minus X to the third power and then you add the words you keep the second function which is 1
over 2 minus X and then to the second power times the derivative of that which is 2x right in my
opinion the roof is kind of boring but then the interesting part that anyways this is going to
be 2x to the 2nd power over that which is 2 minus X to the 3rd power and this is going to be added
with 2x on the top over that 2 minus X to the 2nd power well can we add them up yes but we need a
comedy now first we need to have one more so we multiplied by 2 minus X on the bottom and also on
the top right if you distribute this you get 2x times 2 which is X 2x times negative X which is
negative 2x squared right okay this the bottoms are the same now which is 2 minus X and raised
to the 3rd power combined items on the top this and that cancel each other out you have 4 at left
this right here much cooler isn't it however this kind of things only work if you design
the functions carefully this and that so our parks this answer for you if f of X
is x squared and G of X is 1 over 2 months X and s squared the fake product rule will work