OH SHNIZZLE MORE CALCULUS.
So, now that you have a rough idea of the derivative, let’s ask where it works. Or, to use more formal language, let’s try to figure out what qualities a point needs to have for it took make sense to take a derivative.
I want to try to give you two ways to think about this – first, a graphical way. Second, an intuitive way.
The graphical way is this – recall that when you take the derivative, you’re finding the tangent to the curve at some point. If it doesn’t make sense to take a tangent at some point, it’s probably not differentiable there. The book presents three cases where this is true: corners, jumps, and vertical tangents.
A corner would be a jagged edge. It’s a place where the function suddenly changes. For example, if the function is f(x) = 1 until x=5, then is f(x)=x for larger x, you’ll get this instantaneous change. You can take the derivative before 5 and after 5 quite easily. But at 5, no matter how much you zoom in, there’s always a sudden change. So, you can’t really stick a meaningful tangent line anywhere on 5. Thus, it’s not differentiable there.
A jump is when a function suddenly goes from one point to the other. So, imagine a graph is f(x)=1 until x=10, and f(x)=5 thereafter. Once again, you have a case where at a particular point (in this case, x=10), there is no way to meaningfully assign a tangent line.
A vertical tangent is a little different. This is when you have a portion of a curve in which the tangent line is perpendicular to the x axis. Remember that a tangent is the slope of a curve at a point, and that a curve is “change in y/change in x.” In the case of a completely vertical line, that’s “change in y/0.” If you divide by zero, the universe ends (or your derivative equals infinity, which doesn’t make sense). So, it’s not differentiable there either.
So, you may notice there’s some sort of rule connection differentiability and continuity. And, IN FACT, there is: If a function is differentiable somewhere, it’s also continuous there. This makes sense graphically, but it also makes sense if you recall that a derivative is a special type of limit as you zero in on a certain point.
Of course, as in the case of the vertical tangent, there are situations where the function is continuous despite not being differentiable.
Now, I want to try to give an intuitive sense of what it means to be differentiable. If a function is differentiable over an interval, it’s essentially mimicking reality. So, for example, say you’re jogging. No, wait, that’s unrealistic – say your friend is jogging. From 1pm to 2 pm, he jogs at 3 miles per hour. Every day at 2pm, he changes his speed to 4 miles per hour. If you were to plot a graph of his speed over time, it might very well look graphically like it’s undifferentiable at 2pm, because the graph suddenly jumps. But, if you zoomed in, you’d see that there wasn’t really a sudden jump – there was a very quick curve upward from 3 mph, then an almost (but not quite) vertical line, which plateaued at the level of 4 mph. Why? Because your friend can’t change infinitely fast. In fact, nothing can… well… maybe when we get to quantum mechanics. If he can’t move infinitely fast, you can’t make a graph of him that describes a corner or a jump or a vertical line. All three would require an instant change.
We’ve already gone over this in previous blogs, so I wanna just focus on one aspect. Some students get confused about the way Leibniz notation works with higher derivatives. In the case of higher derivatives, Leibniz notation can be annoying, BUT it’s useful because once again it makes it clear what operation you’re actually doing.
So, for example, when you have and then you take another derivative of y with respect to x, some people expect it to be . If you think this, it’s not just a simple mistake – it means you’re confused about what you’re doing. When you take the derivative with respect to x twice, you’re running the operation “d/dx” on it twice. The y part is just the part being operated on. So it absolutely makes no sense to say that the numerator and denominator are squared. It makes lots of sense to say:
It may not be pretty, but it shows you what you’re doing. It’s actually a good way to tell if students really understand derivatives to ask if they can explain why the above notation makes sense.