So, here we have an example where we are told, given the function \( f(x) \), sketch the derivative of the function \( f'(x) \). Let's see how we can go about solving this. Now, whenever I need to solve these problems where I need to sketch the derivative, I always like to start by figuring out all the places where we're going to touch the x-axis on the derivative graph. And that's going to happen whenever we can draw a line tangent to the curve that is flat, meaning it has a slope of 0. Now, I can see one flat line I could draw, tangent to this curve, would be right there on this peak.
Whenever you have a peak or a valley in your graph, there's going to be a flat line which has a slope of 0. So, that means that right here at negative 1.5, we're going to be touching the x-axis. Next, I can see we have a pretty long portion of the graph where the tangent line would be flat at every single point. So that means from negative one all the way over here to positive 1.5, we're going to have this flat line on our graph that touches the x-axis. So those are going to be the places I see where we have tangent lines with a slope of 0.
There's no other place that I can draw a 0 slope tangent line, so that's going to be all the places we're touching the x-axis. So now, I just need to construct the rest of this graph. Well, if I start from left to right, I'm going to focus on this portion of the graph right here until we get to the 0 tangent line, the 0 slope tangent line, and what I can see is any line that I draw that's tangent to the curve right here is going to have a positive slope. So that means we're going to be above the x-axis at negative 2, and any place that I am on this interval. Next, what I can see is if I look at this next portion of the graph, we'll notice any tangent line I draw here would be negative.
It would slope down. So because of this here, we're going to actually be below the x-axis. So what that does is it allows me to see that this is going to be a straight line, which looks something like this. So we're going to have the straight line. It does have a slope to it because we can see that we have some sort of increasing and decreasing tangent lines, but that is going to be this portion of the graph.
Now, this next portion of the graph right here, we actually already sketched in. But what exactly happens at this point? Well, at that point, notice there's a sharp corner, and whenever we have these sharp corners, that is going to be a jump discontinuity. So that means at this point right down here, we're actually going to put an open circle, because the graph is discontinuous right there for our derivative. Next, I can see that we stop right here at 1.5, and that's another sharp turn that we have.
Because we have another sharp turn that means there's going to be another jump. Now for this last portion of the graph, I can see that we have a line. Now this line does have a certain slope to it, but the tangent line that we draw to this curve, the slope of this tangent line would be constant. So that means that every place here that we see, every place that we see on our derivative graph is just going to be a constant. So what that means we're going to have another line right here which goes forever in this direction, and it's going to be flat because this is a line with a constant slope.
So, this right here is how you can construct the derivative graph, and this would be the solution to this problem. So that is how you can sketch these derivatives when you have these kinds of special cases with sharp corners or flat lines or even lines with a certain slope. I hope you found this video helpful. Thanks for watching, and let's move on.