 Hi and welcome to the screencast where we're going to try to tell where a function is differentiable just given its graph. So recall that a function is said to be differentiable at a point x equals a if the derivative of your function at x equals a exists and that means graphically that if I looked at the graph of my function I should be able to put a tangent line to the graph of this function at x equals a and the tangent line should have a well-defined slope. Another way to interpret that as we discussed in the book is that a function is differentiable at a point x equals a if when you go to the point x equals a and go up to the graph and zoom in on that point then the function's graph ought to appear like a single straight line. The terminology we used in the book was that the function is locally linear at this point. So here's a graph of a function let's call it f it doesn't have a formula it's just a picture and let's look at some points and tell where the function's differentiable where it's locally linear that is and where it is not. Let's start by looking at x equals one and we're going to go to this area on the graph right above x equals one around y equals six and pretty clearly if you zoom in on this portion of the graph it does look locally linear and when you zoom in and you see the straight line appearing the function is not actually straight. There's a bit of a curvature there at x equals one but it looks straight and the more we zoom in on it the straighter it looks. That straight line that you're seeing there is actually what the tangent line would look like if you actually drew it at x equals one. So since the function is locally linear at x equals one then we're going to say that the function is differentiable at x equals one. There are lots of places where this function is differentiable for example at x equals two the function is differentiable because if I zoom in on the graph at x equals two I see something that looks like a straight line a single straight line. So that means that the tangent line could be drawn to the graph at x equals two and that makes the function differentiable there. It might be more helpful to focus on some places on this function where it is not differentiable. Let's start with x equals four. Now what's happening at x equals four is that the function fails to exist there. That's what the hollow circle represents. Now the hollow circle doesn't actually have any physical dimensions to it. But if I were to zoom in on the graph at x equals four, just imagine this circle getting smaller as I zoom in. You would not be looking at a straight line. You'd be looking at a line with a break in it. So that's not a straight line. Therefore, the function's not locally linear at x equals four. And so therefore, the function is not differentiable at x equals four. Again, another way to think about this is if I try to put the tangent line to the graph at x equals four, it wouldn't work. This process is impossible because there's no graph at x equals four to which we can attach this tangent line. So since the tangent line can't be drawn, the derivative fails to exist at x equals four. That means the function's not differentiable there. Another place where the function fails to be differentiable is at x equals six. We can't zoom in terribly far on x equals six because of this jump that the graph makes. And that jump is precisely why the function's not going to be differentiable there. If you were to zoom in on this function, you would certainly not see a single straight line. You would see a couple of line segments perhaps, but not a single unbroken line here. So since I have this jump, that's another place where your function here, is fails to be differentiable, it's not differentiable. Finally, at x equals eight, we don't have a break in the graph here, but we have a corner or a cusp on this graph. The function's going to fail to be differentiable here at x equals eight as well. Again, simply because if I zoom in on the graph at x equals eight, then I do not see a straight line. The function is not locally linear at x equals eight. What I have here is I continue to see a corner, no matter how far I zoom in on the point there on this little v shape. I still see a v shape. I never see the thing actually flatten out like I would if this were just tight curve that I would be making. So this sharp corner there prevents the tangent line from being drawn because the function's not locally linear there. So the function fails to be differentiable at x equals eight as well. So to recap, the function here f is not differentiable at three points, four and six and eight. The function is differentiable everywhere else because if you pick a point other than those three points and zoom in on the graph, the graph will be locally linear upon zooming. So that's a quick conceptual visual idea about how to determine whether a function is differentiable. Thanks for watching.