Justifying the power rule

What I want to do in this video is to see whether the power rule is giving us results that at least seem reasonable. This is by no means a proof of the power rule, but at least we'll feel a little bit more comfortable using it. So let's say that f of x is equal to x. The power rule tells us that f prime of x is going to be equal to what? Well, x is the same thing as x to the first power. So n is implicitly 1 right over here. So we bring the 1 out front. It'll be 1 times x to the 1 minus 1 power. So it's going to be 1 times x to the 0 power. x to the 0 is just 1. So it's just going to be equal to 1. Now, does that makes conceptual sense if we actually try to visualize these functions? So let me actually try to graph these functions. So that's my y-axis. This is my x-axis. And let me graph y equals x. So y is equal to f of x here. So y is equal to x. So it looks something like that. So y is equal to x. Or this is f of x is equal to x, or y is equal to this f of x right over there. Now, actually, let me just call that f of x just to not confuse you. So this right over here is f of x is equal to x that I graphed right over here. y is equal to f of x, which is equal to x. And now, let me graph the derivative. Let me graph f prime of x. That's saying it's 1. That's saying it's 1 for all x. Regardless of what x is, it's going to be equal to 1. Is this consistent with what we know about derivatives and slopes and all the rest? Well, let's look at our function. What is the slope of the tangent line right at this point? Well, right over here, this has slope 1 continuously. Or it has a constant slope of 1. Slope is equal to 1 no matter what x is. It's a line. And for a line, the slope is constant. So over here, the slope is indeed 1. If you go to this point over here, the slope is indeed 1. If you go over here, the slope is indeed 1. So we've got a pretty valid response there. Now, let's try something where the slope might change. So let's say I have g of x is equal to x squared. The power rule tells us that g prime of x would be equal to what? Well, n is equal to 2. So it's going to be 2 times x to the 2 minus 1. Or it's going to be equal to 2 x to the first power. It's going to be equal to 2x. So let's see if this makes a reasonable sense. And I'm going to try to graph this one a little bit more precisely. Let's see how precisely I can graph it. So this is the x-axis, y-axis. Let me mark some stuff off here. So this is 1, 2, 3, 4, 5. This is 1, 2, 3, 4. 1, 2, 3, 4. So g of x. When x is 0, it's 0. When x is 1, it is 1. When x is negative 1, it's 1. When x is 2, it is 4. So that puts us right over there-- 1, 2, 3, 4. Puts us right over there. When x is negative 2, you get to 4. It's a parabola. You've seen this for many years. I put that point a little bit too high. It looks something like this. Actually, the last two points I graphed are a little bit weird. So this might be right over here. So it looks something like this. It looks something like that. And then, when you come over here, it looks something like that. It's symmetric. So I'm trying my best to draw it reasonably. So there you go. That's the graph of g of x. g of x is equal to x squared. Now, let's graph g prime of x or what the power rule is telling us a g prime of x is. So g prime of x is equal to 2x. So that's just a line that goes through the origin of slope 2. So it looks something like that. When x is equal to 1, y is equal to 2. When x is equal to 2, y or g of x is equal to 4. So it looks something like this. Let me try my best to draw a straight line. It looks something like this. Now, does this make sense? Well, if you just eyeball it really fast, if you look at this point right over here-- and you want to think about the slope of the tangent line. Let me do this in a color that pops out a little bit more. So the tangent line would look something like this. So it looks like it has a reasonably high negative slope. Yeah. It's definitely a negative slope, and it's a pretty steep negative slope. For x is equal to negative 2, g prime of negative 2 is equal to 2 times negative 2, which is equal to negative 4. So this is claiming that the slope at this point-- so this right over here is negative 4-- is saying that the slope of this point is negative 4. m is equal to negative 4. That looks about right. It's a fairly steep negative slope. Now, what happens if you go right over here when x is equal to 0? Well, our derivative-- if you say g prime of 0-- is telling us that the slope of our original function, g, at x is equal to 0 is 2 times 0 is 0. Well, does that make sense? Well, if we go to our original parabola, it does indeed make sense. That's the slope of the tangent line. The tangent line looks something like this. We're at a minimum point. We're at the vertex. The slope really does look to be 0. And what if you go right over here to x equals 2, the slope of the tangent line? Well, over here, the tangent line looks something like this. It looks like a fairly steep positive slope. What is our derivative telling us based on the power rule? So this is essentially saying, hey, tell me what the slope of the tangent line for g is when x is equal to 2. Well, we figured it out. It's going to be 2 times x. It's going to be 2 times 2, which is equal to 4. It's telling us that the slope over here is 4. And I'm just using m. m is often the letter used to denote slope. They're saying that the slope of the tangent line there is 4, which seems completely, completely reasonable. So I encourage you to try this out yourself. I encourage you to try to estimate the slopes by calculating, by taking points closer and closer around those points. And you'll see that the power rule really does give you results that actually make sense.