so I'm going to start discussing some Concepts about graphs of derivative functions So based on what we know about functions we can start to come up with some things that we can figure out about what we're going to know about the derivative functions so again back to our definition the derivative of a function at a point x equal a is this limit as H approaches 0 if F of a plus h minus F of a divided by H now F Prime of a the derivative does not exist if F of a does not exist right if your function does not exist at a point we cannot say what is the slope at that point but then more particularly F of a is part of the definition of f Prime of a so if F of a doesn't exist F Prime of a cannot exist all right next statement F Prime of a does not exist if your function has a cusp at x equal a a cusp is just a sharp point in the curve so we're saying if our function has a cusp in it at that point then the derivative cannot exist at that point so here is a function with a cusp right right here is a cusp it is just a sharp point in the Curve now the reason we can't have that is because if I wanted to find the slope right we'd go from a to a plus h and we could find the secant slope but then as we send H to zero this point approaches this point and the slope of our tangent line is that blue line however if I had some other point A Plus H now again we don't necessarily know if H is positive or negative um we're just sending H to zero if I were to connect these two lines I get a secant slope but now if I were to send H to zero this point approaches let's switch colors this point approaches this point which makes our tangent line go this way so the slope of the tangent as we approach from this side and the slope of the tangent as we approach from this side are different which means in this definition as H approaches 0 from the right or as H approaches zero from the left we get two different limits and if the limits from the right and the left are not the same we say the limit does not exist and so because we are saying the limit as H approaches zero whether it is from the right or the left and so that is really the definition of a cusp right this sharp point is sort of just an intuitive definition a cusp actually means as you approach that point who from one side or the other you're approaching a different slope so um the opposite of having a cusp is usually what's referred to as a smooth curve so now if I were to take a point here if I were to approach that point from the left or from the right they're both approaching the same slope so to review F Prime of a doesn't exist if the function doesn't exist F Prime of a doesn't exist if the function has a cusp at that point and then the other thing is f Prime of a doesn't exist if your function is not continuous at that point right if your function is not continuous at x equal a then F Prime of a cannot exist and again we can see this for a similar fashion if I went over here to some value a plus h this is a secant line and then as H approached a this point would get closer and closer to here and again we would get our slope approaching one value now if we went to an A plus h here and we sent H to zero then again our secant slope would be approaching a different slope so this says if F Prime of a exists then the following must be true which is nice so it is sort of the opposite of this if the function doesn't exist F Prime of a can't exist if the function has a cusp F Prime of a can't exist if the function is not continuous and F Prime of a can't exist so therefore if F Prime of a exists that means our function is continuous at that point and smooth at that point which means no cusps and now we say if F Prime of a exists then we say that our function is differentiable at x equal a so the function is differentiable at x equal a that means the derivative exists at x equal a but that implies we have a continuous smooth curve at that point okay so here's a function f and I say sketch F Prime now I did a whole bunch of linear equations so we can see what they are so right here this first segment is a linear equation and the slope of it is 2. so anywhere in here F Prime should equal 2. so I just have a value at 2. now right here the slope is zero so F Prime should just equal zero anywhere here the slope is one which says F Prime should just equal one over here the slope is negative two which says F Prime and that interval should just equal negative two and from here on F Prime the slope is negative one so F Prime should just be negative 1. now the one thing I need to go back and do is for all of these cusps we said the derivative cannot exist at those cusps so we should have a whole bunch of Open Circles where the derivative at those points doesn't exist but for anywhere in between it does right if I wanted to point here right the slope here is one so the output from the derivative should be 1. and so this blue graph would be F Prime of x all right now here is something that we did previously we said that F Prime of X sorry if f of x was the square root of x then F Prime of X is 1 over 2 radical X that's the derivative we found in a previous video now I went ahead and graphed these two things right here's the square root of x now here is one over two radical X this is just a rough sketch but as X gets bigger one over two radical X gets smaller and as X gets smaller one over two radical X gets bigger and again we can only approach Zero from the right because the domain of this function which is zero to Infinity uh but there's a few things to consider just about the shapes of these two graphs for example this function has a positive slope no matter where I go which means that this function has positive outputs no matter where I go right this function f Prime of X does not exist below the x-axis it has only positive outputs because this function has only positive slopes however if we're to look at these slopes even though they're always positive the slopes are getting smaller right and you might start to say that the slopes are tapering off and so if we look over here at F Prime the positive values are getting smaller and the positive values are getting closer and closer and closer to zero because the positive slopes over here are decreasing and so there's a few things that are gonna all start to come together as we consider more things about the properties of a function and the properties of a derivative because the derivative function will tell you the slope of the original function so here's a quick statement of that I just said if a function is increasing that means it's getting bigger a function gets bigger if it has positive slope so if f is increasing F Prime is positive similarly if your function f is decreasing it's getting smaller which means it has negative slope so the next thing to Define is concave down concave down functions look like this but that's what they look like but they have an actual specific meaning right to be concave down means if you were just to look at a slope the slope is decreasing right right here you could have a slope of I don't know say the slope is five and then over here you might say that your slope is three and then maybe the slope is one and then up here maybe your slope is zero and then maybe the slope is negative one so as we go from left to right the slope values are getting smaller and smaller and smaller that is the definition of concave down right the definition of concave down is decreasing slope so if you have a function that's concave down then that means the slope is decreasing which says F Prime of X is decreasing because F Prime of X tells you the slope uh similarly if we have a function that's concave up right it has this shape to it but what concave up actually means is increasing slope maybe right here you have a slope of zero and maybe right here you have a slope of one and maybe over here you have a slope of five the slope is getting bigger and bigger and bigger um the same thing is true over here right maybe this is a slope of negative five and maybe this is a slope of negative one so as you go from negative 5 to negative one to zero to one to five that slope is getting bigger and bigger and bigger so concave up by definition means increasing slope which says then F Prime of X is increasing all right now we have one other thing to just to consider really really quickly and that is f double Prime of x now if we had a function f then F Prime of X tells you the slope of f of x now F double Prime of X the second derivative tells us the slope of f Prime of X so we go back to this last graph really really quickly all right this function f Prime should tell us the slope of this function f however this function right here is still a function it is still a graph it still has a slope so I I can ask questions about this slope which I answered over here but I can also ask you questions about this slope because this is still a function and the slope of this function f Prime questions about that can be answered involving F double Prime the second derivative which tells you information about the slope of your first derivative