# Sketching Derivatives from Graphs of Functions 5 Examples Calculus 1 AB BAM! Mr. Tarrou a little birdie told me
you were having trouble sketching derivative functions from the graph of a
function well my students are struggling with this too and I’m gonna do so I’m
gonna do an extension to a video I did years ago I’ll have a link to that in
the description I think it was this little birdie that told me you were
having trouble and we’re gonna do five more examples of looking at a sketch of
a function and trying to draw a reasonable representation of what the
derivative would look like and remember a drip well actually the fourth the
fifth example were going to work in Reverse we’re gonna look at the graph of
the derivative and try and sketch what the function may look like okay so we’re
gonna look at some settings that are more complicated than I did years ago in
my previous lesson and we have to remember that a derivative tells us the
slope of a function so we’re not looking at just the graphs the function f as a
line or a parabola or a cubic function like I did years ago we’re going to go
into these more complicated settings and we need so we really need to remember
that not just kind of shortcut this process and go well he gave me something
that looks like a parabola so the derivative must be a straight line
because you’re going from a second degree down to a first degree well here
we do have a function that does appear to look parabolic in shape but then when
we get to x equals two it just branches off and we have a
linear function so maybe this is the graph of a piecewise function well what
will the graph of the derivative look like number one I like to try and
identify any points along the function f where the slope therefore the derivative
derivative is giving us a value of zero well certainly if we look at some
tangent lines here and think about the instantaneous rate of change we have a
horizontal tangent line for this function at x equals negative 1 so as we
sketch a reasonable representation of what this derivative should look like
will certainly the derivative which gives a slope and the slope here is you
know horizontal or the function seems to have a horizontal tangent line you
better make sure your derivative goes through the point negative 1 0 now what
happens to the left of X equals negative 1 while the
function is falling and if the function is falling that means that it’s slope is
negative so the derivative to the left of negative 1 needs to be below the
y-axis or the excuse me the x-axis given you a representing that we’re getting
negative values of Y because one more time the answer from a derivative is the
slope and it seems to be helping for me to you know force my kids to not look at
this as one complete picture but break it down into pieces so let’s say we look
here where X is equal to negative 3 you know what does it seem like appear that
you know the slope for this graph here is at x equals negative 3 well if I kind
of just very lightly dot in what might appear to be a good representation of
what a tangent line would be it seems like at least at x equals negative 3
that tangent line has a slope of negative 3 over sort of 1 so at x equals
negative 3 my derivative should be given me a value of negative 3 now you know if
you say the slope there is negative 2 and 1/2 or negative 2 I’m ok with that
we’re looking for a reasonable sketch of what this derivative may look like so
we’re going to be going you know and as we go here at x equals negative 4 the
functions falling faster giving us rip you know which B should be showing a
more negative slope and it’s negative and it’s still falling so the slope is
still negative it’s still falling but not as fast so the slope is less
negative until of course we get to that x equals negative at x equals negative 1
where the slope is 0 and then as we move to the right of negative 1 that function
f starts to rise and of course if a function is increasing that means its
slope is positive and it’s rising just a little bit and then it’s rising a little
bit faster and a little bit faster and so on and so on so that derivative
should be giving us to the right of negative 1
ever-increasing positive values and if there’s some symmetry here and I’ve
drawn this somewhat accurately if I were to visualize a tangent line here at x
equals negative one maybe we could approximate that that slope is well my
drawing is not perfect and maybe that wasn’t quite negative three but maybe
two two and a half of course you know I wanted it does look sort of parabolic in
shapes we’re gonna go ahead and let this derivative the sketch of what a possible
derivative would look like continue to go up in a linear fashion I’m again
while I’m starting to realize it’s helpful to let my kids ask tell me what
the slope is approximately at certain points we kind of you know need this
picture to somewhat look neat of course and we get to this issue here at x
equals negative 2 well it x equals negative 2 our graph really changes
direction and has a sharp bend well don’t forget you can’t find a derivative
at a sharp bend or at a point of discontinuity so as we approach x equals
positive 2 at that exact value of x equals positive 2 at that sharp Bend our
derivative is going to be undefined so you want to indicate with an open dot
that at x equals positive 2 open and say negative 2 earlier that is undefined now
what happens as you go to the right of this sharp bend well we have a linear
function we can clearly see it just up 1 over 1 up 1 over 1 which means that the
slope of this piecewise function the slope to the right of x equals 2 that
stays a consistent value of 1 so I’m going to come down here and remember
that derivative tells us slope I’m going to indicate an answer a y-value of 1 and
oh it’s 1 all the way so our sketch of our derivative should look something to
that effect and that is the sketch of the first derivative now you know do we
have the exact proper you know angle if you will slant on that oblique section
of the grant oblique you know it’s not
perpendicular not vertical I don’t know but it’s a reasonable approximation if a
student you know had a line coming through here and maybe just going up to
a you know a y-value of three instead of my y-value of four four and a half I’m
okay with that but making sure that we are are passing through the x axis at
negative one representing that slope of zero to the left having those negative
slopes to the right having those positive slopes having the open dot
showing us our derivative is undefined at that sharp bend I’m good with that
now if I want to draw another step of that derivative if this is F and this
really indeed is a you know linear function it seems like for this portion
of the graph over here our slope is a little bit less than two over one so the
slope of this derivative this first derivative to the left of x equals two
seems to be consistent let’s say almost two so I’m going to have it just below
that’s Y value of two it’s still on it’s still undefined here and you can’t take
a derivative at a point of discontinuity so I’m going to keep that open dot there
and to the right of x equals 2 we have a horizontal line so the slope will
horizontal line is 0 so our second derivative is going to look like this so
actually this I just did two examples there one over here I got a function f
which has a vertical asymptote at x equals zero and way over here you know x
equals negative 2 4 6 8 ish unless I miscounted it seems like look that graph
is almost horizontal it’s just staying above the x-axis and rising very very
slowly and let’s see I don’t really have a set of markers like it would in my
class but we have a tangent line which has a slope nearly zero we have a
tangent line which is got a slope of I don’t know maybe one ish a little bit
less than one I have a tangent line whose slope is
let’s say maybe positive three I have a tangent line whose slope is maybe
positive ten and that that function that was almost flat having a slope of almost
zero but rising the x-axis is acting as an asymptote starts to shoot off to
infinity as we approach x equals zero so that slope is getting ever and ever
greater approaching infinity so our first derivative here let’s keep it in
orange like our first example you know maybe look something like this I’m gonna
pull it apart just so you can see the difference between the orange and the
white again this is a reasonable sketch of what the derivative may look like I’m
not claiming it to be exotic percent accurate but that function is increasing
giving us of course positive slopes until we go vertical and that slope is
approaching undefined now over here though instead of rising and shooting
off to infinity it’s like as soon as you get past that asymptote
boom it’s falling very very quickly meaning we have very very negative
slopes very very negative so maybe like at 0.5 that slope is like negative 10 or
something like that but that starts to in it’s to the right of this asymptote
that graph never stops falling and if the graph is always decreasing your
derivative is always negative so you better make sure your sketch stays below
the x-axis but you can see it’s flattening out and as you have this
graph flattened out and a start approaching the x-axis as an asymptote
the slope really never becomes zero but it does approach zero and so this is our
sketch of what f prime the first derivative may look like and it you know
just make it a reasonable sketch it is always increasing as X approaches zero
from the left and this function is always decreasing so we have those
negative slopes as you approach 0 from the right so let’s move on to our next
two examples so for our third example we’ve got this sort of wavy function
thing going on here okay well that looks rather complicated
how are we going to sketch oh and this is I didn’t label it but this is some
kind of function not F prime but this is some kind of function and we’re gonna
sketch its derivative well like I’d like to do in my class
let’s highlight it’s kind of if you the easy points if you will you know or the
key points you know like maybe I see a point and I go maybe that slope is
negative you know whatever negative three but it’s really negative two point
five so like that there’s a little bit of room for error but look man there is
a horizontal tangent line right there so at 3 PI over 2 this function it’s slope
is equal to zero so we better make sure that our derivative passes through that
point and like here we have a horizontal tangent line so let’s make sure our
derivative is giving us that answer of zero that slope of zero and as well here
and as well here okay now I kind of cheated here a little bit I did not
include some graph paper grid which get for my handy dandy grid chalk maker
thingy majiggy but at any rate because of the scale being this PI over 2 pi 3
PI over 2 and so on and pi being you know 3.14 approximately yadda yadda
yadda and the vertical scale is 1 so what’s happening well what’s happening
between this horizontal tangent line which has a slope of 0 and this
horizontal tangent line with a slope of 0 well between negative PI over two and
positive PI over 2 this function is rising well the function is rising it
has a positive slope therefore the derivative have better e giving you
positive answers and where between negative PI over two and positive PI
over two is this function rising the fastest where does it seem like it’s
giving us the biggest answer for the slope well it’s it’s right there because
and you know forgive my drawing I’m not a computer but it’s rising slow
just a little bit to the right of negative PI over two and it’s rising
quicker and quicker and quicker and we have sort of this greatest icy sort of
like a most steep or the steepest slope here at x equals zero and then it kind
of starts to come over well that’s because we’re going from concave up to
concave down at x equals zero we actually the point of inflection so at
that steepest visual slope let’s make sure that our derivative is giving us
the largest possible answer and it doesn’t really look like a slope of one
but I kind of happen to know that that one is so but between negative PI over
two and positive PI over two it’s going it’s right it’s has a slope of zero
rising slowly and faster and faster and faster and then that slope starts to
decrease so our derivative is going to look something like this but I think
someone’s at my door let me come right back
alrighty then so to finish this up yes zero zero and rising all the way having
that derivative given us positive values between this x value of zero slope of
zero and between this slope of zero our function is falling and like over here
instead of rising its fastest rate we have its most negative slope so this at
this point here where we go from being concave down to concave up our
derivative is giving us our smallest value and I just happen to know that
that’s going to be equal to negative one and so as we kind of connect the dots Oh
nope our functions rising again and so on and so on
we get zero and zero our slopes are function falling from negative 3 PI over
2 to negative PI over 2 and here it’s falling its fastest rate so falling
means slopes which are negative we get the graph of our first
derivative so this is our F prime of X now did you notice a pattern between the
white graph and the orange graph as I was drawing that I hope so because that
means you know your unit circle this is f of X as it turns out happens to be
equal to the sine of X and this orange graph happens to be well what does it
look like at x equals 0 the function is giving us a value 1 of 1 at PI over 2
that function is giving us a value of 0 at PI this function is giving us a value
of negative 1 actually this derivative function in case you never really
connected dots before and that’s not really drawn quick let me just change
that a little bit is the cosine of X right the derivative of sine is cosine
and there you can see it graphically over here on our fourth example we have some kind of roots function and this
function at negative 2 4 6 you know 8 or so negative 9 whatever but we have an
x-axis here that is acting as a horizontal asymptote again like our
previous example before and this function is rising but very very slowly
so back here at I counted this but it’s now let’s bother me I forgot it was
negative 2 negative 4 negative 6 negative 8 at negative 9 our slope is
just barely positive this this function is increasing but very very slowly well
okay and it’s increasing a little bit faster and it’s increasing a little bit
faster giving us more and more positive slopes maybe here maybe the slope here
is equal to positive 1 but then it goes very vertical between the x value of
negative 1 and the x value of 0 and actually I’m just trying to draw this as
if there was a vertical tangent line at x equals 0 so the graph of this derivative from this function
f is barely increasing increasing faster
and faster and really going vertical saying that our slope is of course the
slope of a vertical line is it’s not really it’s not an asymptote like our
previous example but the slope of a vertical line is undefined we’re
approaching a slope of infinity right you cannot find a derivative you can’t
find the slope of a vertical line or you can it’s sort of it’s you can say it’s
undefined and then as just as you pass the other side of x equals 0 you get to
the right of that vertical tangent line that graph is still increasing very very
rapidly and then slows down and starts to rise very very slowly meaning the the
slope right this is concave down and when a function is concave down that
means the first derivative is decreasing but this function while sorts two starts
to layover and act like there’s a horizontal asymptote out here somewhere
this function is never going to stop increasing it just increases very very
slowly so just to the right very large values for the slope we need to as far
as the derivative is concerned see that there is a vertical asymptote at x
equals zero there’s not an asymptote for the original function but for the
derivative and then we come down here and make it look something like that
that is a reasonable that orange line that orange graph is a reasonable sketch
of what the first derivative would be for this function and again I want to
focus on the graph of the function not necessarily giving you the equation of
the function how having you find algebraically the derivative of that
function and then graphing it with a graphing calculator wants you to see it
and draw that reasonable sketch from a graph and just using your knowledge you
know of the fact that a derivative is just simply giving you that slope I got
one more example coming up right now okay so we have you know I’d basically
draw a parabola and but this time I wrote in white but then I realized all
my derivatives were in orange for the previous four examples so let’s be
consistent I have a derivative so we’re going to look at the graph of this
derivative and try and sketch what would be a reasonable approximation of what
the antiderivative or basically function f is going to look like well one more
time right derivatives give us slope and what is this derivative telling us about
the original function well it’s telling us that x equals negative three and at x
equals M positive three make a big ol dot kind of blur that over a little bit
that the slope mean a derivative tells a slope and the answer we’re getting from
this derivative is zero so our original function on ok.now may be well I could
ever function go through there except that I don’t want you think that the
values from the derivative is that are the values from the function so if you
know the original function could go there f could go through there but we
better make sure that our function that we’re about to draw has a horizontal
tangent line you know has a slope of zero at negative 3 and positive 3
now what’s happening between so you kind of wanted to mark these is sort of like
important points so let me do this here we go
between those two markers our derivative is obviously giving us negative values
which means that between negative 3 and positive 3 our function that we’re about
to draw ahead better be what decreasing because our slopes are that we’re
getting from the derivative is negative so I don’t know here so we have want to
kind of visualize a horizontal tangent line
and I’m going to visualize a horizontal tangent line there and I just hit my mic
so I hope that static sound wasn’t too bad on the mic between here and here my
function needs to be falling we need to have horizontal tangent lines and
actually if we can approximate a tangent line who has a slope of negative 3 that
would be even better so if I maybe here 1 2 3 okay some falling if I want to
drop that down a little warm again it’s just a reasonable sketch but so once for
a period of time to have a tangent line which looks like it sort of got that
negative 3 slope but zero slope zero slope most negative slope in the middle
and all negative values from the tangent excuse me all negative values from the
derivative so our function needs to be falling something of that effect now
what happens to the left of negative 3 well the answers that we are getting
from our derivative are all positive and if my derivative is giving us positive
answers then our function needs to be increasing right if the function is
increasing our derivative is giving us slopes which are positive and it’s got
back here a very large value from the derivative
so a very positive slope very fast increase or increasing function and it’s
gonna look something like this a little bit positive more positive more positive
more positive more positive as far as its increasing so it’s increasing slowly
its increasing faster course that seems weird because I’m doing this in Reverse
because I just want to make a smooth graph it’s increasing very very fast so
let’s let’s try and visualize this again so we have a function that’s increasing
and increasing very steeply giving us very positive values of slope and it’s
increasing a little slower chalk down it’s increasing a little
it’s not increasing the slope is zero and then to the right of X equals three
we go from a slope of zero now this is going to kind of flow a little better
because I’m going off to the right of the graph but slope of zero horizontal
tangent line slope of positive one so it’s starting to whoops that’s your
orange starting to swing up and that those slopes that we are getting are
getting ever bigger as x increases so as we swing to the right this function
needs to be increasing you know faster and faster and faster because the slopes
we are getting from the derivative are you know greater and greater so this
would be maybe a good approximation of what F looks like now that’s not the
only good approximation for the graph of F by the way you know when you take the
derivative of a cubic function you get a function with the degree of two so I’ve
drawn what looks like a cubic function a third degree and indeed the the
derivative appears to be sort of like a second degree parabola but what is the
if I say f of X is equal to three and I asked for f prime of x or say if I just
say well I have a value of three what’s the derivative with respect to X of 3
well that’s going to be zero right and when you take a derivative of a constant
it just kind of goes away so when we learn we haven’t learned it yet but when
we learn how to integrate these constants are effectively like vertical
shifts like way back in precalc if I say that we have f of X is some kind of
function sketch f of X plus 3 you would just take that original parent function
and shift it up three units well when you take the derivative of that constant
it’s zero when you go to integrate that’s like there’s nothing in there so
what I’m trying to get across is I could basically now take this sketch of my
cubic function and just move it up and down anywhere I like and it would still
be a reasonable sketch because if there was a vertical shift after you
take a derivative it goes away you don’t know what that vertical shift is in
other words I can say hey at x equals negative 3 make sure I have a horizontal
tangent line at x equals positive 3 make sure that slope is 0 make sure that
you’re visualizing kind of like a horizontal tangent line
hey the derivative is negative between these two values so between negative 3
and positive 3 make sure your function is falling to the left of negative 3
your derivative is giving you positive values so make sure your function is
increasing to the right of 3 our derivative is giving us positive values
so make sure to the right of 3 your function that you’re trying to sketch f
is rising either one of those sketches of what may be or anti derivative of F
prime of X you know effectively just an idea of what F may look like is
acceptable because you know through the derivative process we lose that plus
constant we lose that vertical shift I hope this helps you now understand how
to take the sketch of a function or even its a derivative and come up with a
sketch of say a second derivative or in this case the antiderivative function f
from f prime so it’s a challenging I hope this helps you see how to give
these sketches get these sketches done with a little bit more accuracy I miss
true ma’am go to your homework

### 12 thoughts on “Sketching Derivatives from Graphs of Functions 5 Examples Calculus 1 AB”

• October 25, 2017 at 2:27 am

Awesomeness

• October 25, 2017 at 3:46 am

I just wanna say this,”You are a Legend!!!”

• October 25, 2017 at 10:11 pm

I know this is a calculus topic, but which trigonometry book does he teach from?

• October 25, 2017 at 10:11 pm

Liked,

• October 26, 2017 at 3:29 am

Perfect timing got a test friday.

• November 2, 2017 at 4:28 am

@ProfRobBob could you possibly make a video on inverse Trig identities? You made one before, but you did not cover a lot. Could you make a video with more examples like tan inverse (sin 5pie/2)? Example like that? and maybe with with sec, csc, and cot? It would mean a lot! I have a test in about a week, and it would be so helpful to have a video out by thiss weekend by 4 of Nov!

• November 8, 2017 at 1:37 pm

got an exam tomorrow, I'm having a marathon of your videos! Thanks a lot for being my lifesaver all through my high school years!

• November 20, 2017 at 2:11 pm

TECHNIQUES ON HOW TO SOLVE MULTIPLE INTERGRALS
SEE THE VIDEO

• September 23, 2018 at 2:06 am

Chalk down!

• February 18, 2019 at 12:25 pm

MR ben?

• March 18, 2019 at 12:36 am
• 