Derivative as a function
Connecting f and f' graphically
I have a function f of x here, and I want to think about which of these curves could represent f prime of x, could represent the derivative of f of x. Well, to think about that, we just have to think about, well, what is a slope of the tangent line doing at each point of f of x and see if this corresponds to that slope, if the value of these functions correspond to that slope. So we can see when x is equal to negative 4, the slope of the tangent line is essentially vertical. So you could say it's not really defined there. But as we go slightly to the right of x equals negative 4, we just have a very, very, very positive slope. So you could kind of view it as our slope is going from infinity to very, very positive to a little bit less positive to a little bit less positive, to a little bit less positive, to a little bit less positive. So which of these graphs here have that property? Remember, this is trying to graph the slope. So which of these functions down here, which of these graphs, have a value that is essentially kind of approaching infinity when x is equal to negative 4, and then it gets less and less and less positive as x goes to 0? So this one, it looks like it's coming from negative infinity, and it's getting less and less and less negative. So that doesn't seem to meet our constraints. This one looks like it is coming from positive infinity, and it's getting less and less and less positive, so that seems to be OK. This has the same property. It's getting less and less and less positive. This one right over here starts very negative and gets less and less and less negative. So we can rule that out. Now let's think about what happens when x gets to 0. When x gets to 0, the tangent line is horizontal. We're at a maximum point of this curve right over here. The slope of a horizontal line is 0. Remember, we're trying to look for which one of these curves represent the value of that slope. So which one of these curves hit 0 when x is equal to 0? Well, this one doesn't. So the only candidate that we have left is this one, and this one does hit 0 when x equals 0. And let's see if it keeps satisfying what we need for f prime of x. So after that point, it should start getting more and more negative. The slope should get more and more and more negative, essentially approaching negative infinity as x approaches 4. And we see that here. The value of this function is getting more and more negative, and it's approaching negative infinity as x approaches 4. So we'll go with this one. This looks like a pretty good candidate for f prime of x.