the dy and then the Delta Y they're just off by a little bit
okay this video I'm gonna discuss the difference between Delta Y and dy and to do this I will just
begin by drawing a picture for you guys and let me also pick a specific function to work with for
demonstration purpose let me just use square root of x because this is not too easy you are not so
typical and then from here I will pick a point but I'm not going to use 0 0 no 1 1 because they're
too boring let me pick 4 comma 2 because when X is equal to 4 the wife is nice to equal to 2 so
you know it's a nice point to work with now I will pick another point and for this right here I will
pick easy point I'm going to choose 9 for the X when X is 9 you see that the Y value is going to
be square root of 9 and that's precisely equal to 3 so that's nice right and from this picture you
see we have 2 points this was our original point we won from this point to here and then we went
up to get to the new point this right here it's precisely the change in the X values and we can
put down the Delta right here to stand for change this triangle that's the Delta stands for change
and similarly we went up from 2 to 3 and this right here will be the Delta Y and based on this
picture I can do some computation for you guys Delta let's do tota Y first Delta Y we went from 2
to 3 so you can just do 3 minus 2 that changes one but let me actually write down square root of 9
minus square root of 4 right because I plug in the X values into the function to get the Y values and
then this give me a nice number 3 and this is 2 so in this case the author Y is 1 and that's nice and
the Delta X as we can see we went from 4 to 9 so that's just 9 minus 4 and that's equal to 5 and 1
this right here was so easy because I chose nine on purpose to just make everything easy
and that's it what if I didn't pick nine by instead maybe I paid five maybe I paid $700
picture of course and I still have the four comma two and let me just say I am going to pick the x
value be four point five let's pick five yeah just fine nope or four-point match is five so
I have this new x value yes you can plug in five into square root and you can get the Y value and
this Y value right here now it's going to be square root of five well we went up from two
to square root of five in this case you see this is the Delta X okay computer.easy but how about
the Delta Y we already Stan Delta Y is square root of 5 minus square root of 4 but seems like
I'm stuck because I don't have a calculator with me okay but you know that's food on something so
we can feel that we are making progress Delta X this is just 5 minus 4 so this is easy and you
see remember X is the independent variable you get to choose why I've actually won and please
don't comment something that what if I choose X to be PI or square root of 57 or something
please just pick some thanks x values right here because our focus is on getting the change in the
Y values okay all right ideally speaking you want to have the Delta X and Delta Y because
they are the exact change and then you just put this down for you guys I mentioned that the old
house stands for change right Delta Y this right here is the exact and the keyword right here is
the exact change in the Y values and of course want to figure out what if we move around along
the x-axis what happens to the Y values and yeah ideally if I can find it exciting that would be
perfect but right here I have some trouble as I said we are going to be using calculus
dy and also the DX which is stands for the first derivative and one of the interpretation for that
is the slope of the tangent line therefore why don't we draw the tangent line keep in
mind this was our given point 4 comma 2 let's go ahead and draw the tangent line right here
and of course the slope of the tangent line right here is the derivative I will just put down dy DX
okay this is the slope and now you will see okay if I pick X is equal to 4 plug into the tangent
line equation I want to have two because they touch each other f 4 comma 2 and if I plug in 5
into the tension equation I will have this point right here and notice though this point in black
versus this point in red it's just slightly off right it's just off by a little bit this point
in black this point red it seems like this point in red is slightly higher than the point in black
but the good thing about this is that we all for this part right here I can still draw the similar
picture like a little triangle and this part is the DX part and this part here is the dy part
so you see that originally we're trying to get Delta Y which is this distance it was in black
right here okay but we cannot do that it's okay because I will show you how can actually compute
the Y and now you may be wondering what exactly is dy first it has a special name this is called
a differential but more importantly you have to remember the following dy it's also a change but
this right here is just approximated change in the Y values and the key right here is by using
derivative of course because we are talking about the flow of the tangent line so this is the main
difference Delta Y is the exact change and dy is just the approximated change for the actual change
and to do dy you have to use the relative that's the main key okay so right here of course we have
to talk about the rough D now right so here we have the equation I will just put this down again
why it's equal to square root of x let's do that the rough day on both sides so you on the left
hand side I will put down dy DX and the derivative of square root of x is 1 over 2 square root of x
and now on the left hand side if you look at this as a fraction of dy over DX a ratio of dy and DX
let's multiply the x on both sides you will see D Y is equal to 1 over 2 squared of X DX and this
right here is the so-called differential and we use differentials to do approximation for the
actual change and this right here is just pretty much derivative and here is the deal from here I'm
going to tell you the x value that we care which was the four that we have so X is equal to 4 and
how about the DX though because remember I'm doing this times that XS what because that's what's the
original with choose attention line at X is equal to four so that would be the slope of the tangent
line the DX is this change from four to five and we are just taking the DX to be the same as that
your tax so we take DX to be Delta X in your case is the same as one and then from here of
course we can plastics in dy is equal to one over two times square root of 4 times the DX which is
one like this and we can just work this out real quick this is 1 times 1 which is 1 over 2 times
2 which is 4 and this is 0.25 so Y is actually is this debt this is telling me that okay originally
the Y value was here too but we went up by 0.25 so this is 2.25 and combining with everything was
saying we're just trying to do the approximation you see square root of 5 is approximately the same
as when you went up from 2 this is the original Y value and then we added with that dy in the
0.25 so I will just put this down this was the original and then this right here is the dy and
we are saying square root of 5 is approximately 2.2 5 and you see earlier I couldn't get Delta Y
because I cannot do square root of 5 but it's ok because with this I can figure out dy and
this is actually a pretty cool approximation the DX cannot be too big meaning the distance
between the original point and the new point the difference between the X values cannot be too big
you
so I will say this is the pretty cool approximation anyway this is
it hopefully it gets all like this video and if you like my video and
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