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Derivative as slope of curve

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Sal solves a couple of problems where he interprets the derivative of a function at a point as the slope of the curve, or of the line tangent to the curve, at that point.

Video transcript

- [Voiceover] What I wanna do in this video is a few examples that test our intuition of the derivative as a rate of change or the steepness of a curve or the slope of a curve or the slope of a tangent line of a curve depending on how you actually want to think about it. So here it says F prime of five so this notation, prime this is another way of saying well what's the derivative let's estimate the derivative of our function at five. And when we say F prime of five this is the slope slope of tangent line tangent line at five or you could view it as the you could view it as the rate of change of Y with respect to X which is really how we define slope respect to X of our function F. So let's think about that a little bit. We see they put the point the point five comma F of five right over here and so if we want to estimate the slope of the tangent line if we want to estimate the steepness of this curve we could try to draw a line that is tangent right at that point. So let me see if I can do that. So if I were to draw a line starting there if I just wanted to make a tangent it looks like it would do something like that. Right at that point that looks to be about how steep that curve is now what makes this an interesting thing in non-linear is that it's constantly changing the steepness it's very low here and it gets steeper and steeper and steeper as we move to the right for larger and larger X values. But if we look at the point in question when X is equal to five remember F prime of five would be if you were estimating it this would be the slope of this line here. And the slope of this line it looks like for every time we move one in the X direction we're moving two in the Y direction. Delta Y is equal to two when delta X is equal to one. So our change in Y with respect to X at least for this tangent line here which would represent our change in Y with respect to X right at that point is going to be equal to two over one, or two. And it's almost estimated, but all of these are way off. Having a negative two derivative would mean that as we increase our X our Y is decreasing. So if our curve looks something like this we would have a slope of negative two. If having slopes in this a positive of point one that would be very flat something down here we might have a slope closer to point one. Negative point one that might be closer on this side now we're sloping but very close to flat. A slope of zero, that would be right over here at the bottom where right at that moment as we change X Y is not increasing or decreasing the slope of the tangent line right at that bottom point would have a slope of zero. So I feel really good about that response. Let's do one more of these. So alright, so they're telling us to compare the derivative of G at four to the derivative of G at six and which of these is greater and like always, pause the video and see if you can figure this out. Well this is just an exercise let's see if we were to if we were to make a line that indicates the slope there you can do this as a tangent line let me try to do that. So now that wouldn't, that doesn't do a good job so right over here at that looks like a I think I can do a better job than that no that's too shallow to see not shallow's not the word, that's too flat. So let me try to really okay, that looks pretty good. So that line that I just drew seems to be indicative of the rate of change of Y with respect to X or the slope of that curve or that line you can view it as a tangent line so we could think about what its slope is going to be and then if we go further down over here this one is, it looks like it is steeper but in the negative direction so it looks like it is steeper for sure but it's in the negative direction. As we increase, think of it this way as we increase X one here it looks like we are decreasing Y by about one. So it looks like G prime of four G prime of four, the derivative when X is equal to four is approximately, I'm estimating it negative one while the derivative here when we increase X if we increase X by if we increase X by one it looks like we're decreasing Y by close to three so G prime of six looks like it's closer to negative three. So which one of these is larger? Well, this one is less negative so it's going to be greater than the other one and you could have done this intuitively if you just look at the curve this is some type of a sinusoid here you have right over here the curve is flat you have right at that moment you have no change in Y with respect to X then it starts to decrease then it decreases at an even faster rate then it decreases at a faster rate then it starts, it's still decreasing but it's decreasing at slower and slower rates decreasing at slower rates and right at that moment you have your slope of your tangent line is zero then it starts to increase, increase, so on and so forth and it just keeps happening over and over again. So you can also think about this in a more intuitive way.