When you�re studying derivative one of the popular problem that give students fits a
lot of times is you will be given a graph of f(x) but not told what it was you just
be provided a graph like this blue graph right here and they will ask you to graph the derivative
of F without really knowing what F is written out algebraically just from the property of
the graph. So here some hints and somethings to look for when you trying to work these
problem. The biggest hint is how you get started with the problem and the hint is look for
the max and the min of your function. Because if you think about it look at this point right
here it a relative maximum or a local maximum for F and we would know right here the derivative
equal 0. So right here if the derivative equal 0 then that would make it a route of the derivative.
So these give you some solid steady consistent point to plot for any type of function, so
we will put a dot here and the yellow is going to be F prime or the graph of that prime and
right here it would have f prime will have another route F prime is 0 right here as well.
So those are going to be your rock solid points and we are going to base the rest of this
off of. Now to get the rest of the graph to figure out what it�s doing in between these
point here before and the middle and after these two yellow dots let�s look at the
graph here of F so you notice before this first yellow point do you see how this graph
is increasing it�s going upwards now what does that infer about the derivative. Well
if your graph is increasing I think that is equivalent to saying that F prime is positive
you have positive slope so you�re original function is increasing. So if your derivative
is positive that mean it just needs to be hanging out above the x-axis it need to have
positive Y values I did not say that the derivative slope has to be positive just the derivative
Y-values themselves has to be positive. So back here I need to be hanging out up here
not down here, positive slopes. Between the yellow dots I switch colors here to green
we see that these slope for the original function are negative so we are decreasing we know
if these first derivative is less than 0 so you have negative slopes the function is decreasing,
so when I draw the derivative graph I need to be down here in between these yellow dots
not up here. Likewise for the last interval after the second dot the function is increasing
again which means that the first derivative would be positive again. So putting it altogether
how can we stay above the X-axis come down touch this yellow dot go below the X-axis
to get negative Y-values come back touch this yellow dot and they�ll be positive after
that. Well if you put it altogether I think your graph will look something like this positive
Y-values you hit the first derivative equal 0 you hit the route for the first derivative
and then it comes down but than it once you�re negative you�re going to have to turn around
some point to get back to this other route and that after that route you go back up again.
So this is the graph of F prime where this is the graph of F. Now In hind sight let check
a few things, before this yellow dot the derivative is positive and notice it starts off the derivative
at a very high slope but as you move into this local maximum don�t this derivative
get lower and lower and lower and kind of mesh into or converge into 0 but sure enough
your slope do starts off high and then the slope get lower and lower and lower. Now what�s
confusing is I�m not looking at the slope of F-prime I�m only looking at the Y-values
alone at F-prime. Now one last little thing to convince you that, that graph that I drew
is correct if you look at this blue graph I�ll write this in blue this kind of look
like a cubic if you�re familiar with just your basic quadratic and cubic and linear
function this goes up down up that just looks like a cubic. Now what will the derivative
of a cubic function will be something that has an x to the third probably something with
an xsquare well look at the orange graph the graph with the derivative doesn�t that look
like a quadratic? It does, so that farther explain or show that this really is the graph
of F-prime. Now we are going to finish up with a slightly harder example we�re going
to go thru this one a little bit quicker here�s the graph of that of F but in two color I�m
going to do one in red and one in yellow I�m going to try to draw the graph so that F-prime
and F-double prime, let�s do F-prime first so what I know what I need to do is start
off looking for the max and min all these places, these three places the derivative
is going to be 0 so I�m going to go head and put those on the X-axis and those are
what you might call the pivot point or the stationary points or what happen these are
the route of the derivative F-prime is 0 here, here, and here. Now before this red dot the
original function is decreasing, so that means the first derivative would be negative so
I need to hang out below the X-axis. Between this red dot and this red dot the original
function is increasing so the derivative need to stay above the X-axis. So I�m below the
X-axis and then I�m above the X-axis now I have to get back to this route so I�m
going to have to turn around at some point and come back touch this route between this
red dot and this red dot the original function is decreasing its going down so my first derivative
needs to go below the X-axis and then somewhere between the red dot I�m going to have to
turn around to get back to this route which I do, and then after this red dot let�s
look at the original again, now the graph is increasing so I need to have positive slopes
again. So that red right here would be F-prime. Last step we are going to look for the second
derivative of F which is as you guys well know is the first derivative of the first
derivative, so we can just do this process again just looking at the red graph so completely
ignore the blue graph we�re just going to be looking at the red graph. Red graph has
derivative of 0 here and here so I�m going to put a dot here and dot here and then the
red graph before this yellow dot is increasing so I need to hang out above this X-axis and
then between the yellow dots the red graph is decreasing, it�s going down so the second
derivative need to be negative. Then you have to come back and touch this other yellow dot,
now after this yellow dot the first derivative is increasing again so the derivative of that
will be positive in which case on the second derivative graph you will be increasing again.
So we have the original the first and the second derivative. Now in closing I want to
highlight something very, very interesting I�ll do this in green look at the root of
the second derivative right here it changes from concave up now the second derivative
it go from positive to negative right here this yellow dot which means the original function
changes from concave up to concave down, I�m not saying the second derivative changes from
concave up to down saying the original changes from concave up to down right here. So if
you this follow up and look on the blue graph look at that this point right here what would
you call that you go from concave up in which case you have a positive second derivative
to after that point its concave down that mean that this is a point of inflection. And
sure enough we see that took place for the second derivative is 0. Now if you follow
this blue graph its concave down and where does it turn concave up it looks like right
about here it look like that a point of inflection we follow it down that�s a root of the yellow
graph of the second derivative. So all these things fit together. Now what about degrees
(let me color code these degrees) the original looks like probably a degrees for polynomial
it goes down, up, down, up. And then the first derivative looks like a cubic equation degree
3 up, down, up. And then how about the yellow graph look like a quadratic of degree 2. So
all these things fit together so it looks like this is correct. So these can be very,
very confusing I will encourage you to practice these a lot, do lots and lots of examples
because once you get the hang of it, it is easier but it takes a lot of time just thinking
about these things and really try to wrap your mind about how the original, the first
and the second derivative graph are all interrelated.