Differential Calculus (2017 edition)
More practice with the relationship between the graph of a function and the graph of its derivative. Created by Sal Khan.
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- Is it true that if you keep on taking the derivatives of a formula, and graphing the derivative, and then the derivative of that, and then the derivative of that, of a wavy line, the amplitude will continue to decrease and the frequency will continue to increase?(15 votes)
- The effect of repeatedly taking derivatives depends on the nature of the function. Repeatedly taking derivatives of polynomials wipes them out, as the highest order term keeps getting reduced until it becomes a constant and then zero. Repeated derivatives of sine or cosine wrap around in a cycle every four times: sin, cos, -sin, -cos, and back to sin. For hyperbolic sine and cosine the cycle is just two: the derivative of sinh is cosh, and the derivative of cosh is sinh. Most impressive of all, e^x is its own derivative, so that no matter how many times you take the derivative you keep getting the same thing, e^x.(43 votes)
- Sorry that was bad phrasing, my english is not very good. Let me rephrase it. I noticed that when the derivative of y=f'(x) is graphed, it has the same x intercepts as y=f(x). Is this a coincidence or is it a theorem? Can anyone please explain or link me? Thanks! :)(9 votes)
- Here, it's actually just a coincidence. When the second derivative (derivative of the derivative) touches the x-axis, the derivative of the function usually goes from decreasing to increasing or vice versa. In this graph, that just seems to happen at the x-intercepts of f(x). Since at the x-intercepts of f(x), the graph's speed goes from increasing to decreasing or vice versa, the x-intercepts of f(x) and the x-intercepts of the second derivative of f(x) are the same. This is just a coincedence, though, and won't always happen.
You'll learn more about this when you reach concavity here:
I hope this helps!(10 votes)
- Has anyone clearly understood why the green function cannot be f(x) and yellow function, f'(x)?
I think that I repeatedly played the video, but my confusion is not cleared.
- for a function to be f'(x) it should denote the slope of f(x) at that value of x . Consider the part to the extreme left. The slope of the green function is positive but at that value of x the value of the yellow function is negative. So The yellow function cannot be f'(x) . Now, the slope of the yellow function is clearly depicted by the green function so the green function has to be f'(x) and f(x) has to be the yellow function(3 votes)
- at2:20Sal says that the derivative of f(x) should be going downwards or decreasing how did he know that from "eyeballing" it?
I thought the derivative would be positive whenever the function was sloping upwards.u.. but I guess not :((3 votes)
- Be careful.
You are correct that f '(x) is positive for the whole left side of f(x) where the slope is positive (so say for x values -10 to -4). But f '(x) is only "increasing" from -10 to around -7.5. It's difficult to see on the curve for f(x) but obvious looking at f '(x).
If you were to get the slope of f(x) at the far left it would be increasing 1, 2, 3 and peaking at 4 around x = -7.5. Then decreasing to 3, 2, 1, 0. Note that those decreasing values 3, 2, 1 are still positive. Again its not easy to see just looking at f(x) but the graph of f '(x) makes it clear.
Hope that helps!(2 votes)
- Sal, where can I get the derivative intuition module?(2 votes)
- If you equate the derivative to zero, does that mean that the function would change direction?(2 votes)
If the point in question where the derivative is 0 also happens to be a max or min point, then yes, the function would change direction. If it is not a max or min point, then it is an inflection point and what is happening is the function is changing curvature. Here is an example:
- Is the slope of the tangent line (derivative) of a point on f(x), the the value of the point on the graph of f '(x) or is it the slope at that point on f '(x)?(2 votes)
- How can you say that the slope of the tangent is 2-half?(2 votes)
- Sal is just estimating here. Without knowing the equation here it is impossible to say what the tangent sloe of f(-9) is, but if you just draw a line that looks "close enough" to tangent at that point you can guess a slope. Sals line looks pretty close and is about 2.5 in slope(2 votes)
- 3:00How can a slope be positive while it is decreasing?(2 votes)
- At1:01, Sal says that the orange function's slope is not positive at that point, therefore it cannot be the green functions derivative. It looks positive to me, am I missing something?(1 vote)
- The orange function IS the slope. So you are looking for the VALUE of the orange function, not the slope of the orange function. Since it is below the x-axis, it's value is negative.(3 votes)
So what we have plotted here is some function-- let's call it f of x-- and its derivative, f prime of x. And what we need to figure out is which one is f of x and which one is f prime of x? So let's take a stab at it. Let's think about what would be the situation if the green function were f of x. So let's see if this works out. If the green function were f of x, does the orange function here, or the yellow function, could that be f prime of x? So let's think about what's happening to this green function at different points. So this green function right over here, right at this point if we start at the left, has a positive slope. If this orange function were f prime of x, if it were the derivative of the green function, then it would have to be positive because the green function's slope is positive at that point. But we see that it's not positive. So it's pretty clear that the green function cannot be f of x, and the yellow function cannot be its derivative, because if this was its derivative, it would be positive here. So that quickly, we found out that that can't be the case. But let's see if it could work out the other way. So it's starting to feel-- just ruling that situation out-- that maybe that this is f of x and the green function is f prime of x. So let's see if this holds up to scrutiny. So what we have when we start off at the left, f of x, or what we think is f of x, has a reasonably positive slope. Is that consistent? Well, yeah, sure. Our green function is positive. In fact, at the point, it's telling us that the slope of the tangent line is around 2 and 1/2. And it actually does look like the slope of the tangent line is exactly 2 and 1/2 of this function right here. Actually, let me erase this, just so we don't look like we're trying to take the slope of the tangent line of the derivative. So it looks just like that. So we see the slope of the tangent line right over here looks like about 2 and 1/2, and the value of this function up here looks like it's about 2 and a 1/2. So, so far this green function looks like a pretty good candidate for the derivative of this yellow function. But let's keep going here. So let's think about what happens as we move to the right. So here, let's see. It looks like the slope of this yellow function-- let me just use a color we can see-- it keeps going up. It keeps going up, keeps going up. And then at some point, it reaches some maximum slope, and then it starts to go down again. The slope starts to go down again, all the way to the slope going all the way down to 0 right over here. Well, does this green function describe that? Well, let's see. The slope is positive and increasing up to this point, which seemed pretty consistent with what we just experienced. Then the slope stays positive, but it's positive and decreasing. And that's what we saw here. The slope is positive and decreasing all the way to the slope getting to 0 at this maximum point here. And we see, indeed, that on this green function, the green function hits 0. So it seems like it's doing a pretty good job of plotting the slope of the tangent line of the orange function. And then our slope becomes more and more negative. And then it hits some point, some minimum point right over here. The slope hits some minimum point right over here. And then it becomes less and less negative. Let's see how well I can draw it. The slope becomes less and less negative until it hits a 0 slope again. And then it starts becoming positive until it hits some maximum slope. But then it stays positive, but it becomes less positive. It becomes less and less positive. So it looks pretty clear that the orange function is f of x, and the green function is f prime of x.