Main content

### Course: AP®︎/College Calculus BC > Unit 2

Lesson 1: Defining average and instantaneous rates of change at a point- Newton, Leibniz, and Usain Bolt
- Derivative as a concept
- Secant lines & average rate of change
- Secant lines & average rate of change
- Derivative notation review
- Derivative as slope of curve
- Derivative as slope of curve
- The derivative & tangent line equations
- The derivative & tangent line equations

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Derivative as slope of curve

Explore how to interpret the derivative of a function at a specific point as the curve's slope or the tangent line's slope at that point. Dive into problem-solving examples that demonstrate this concept and strengthen your understanding of derivatives in calculus.

## Want to join the conversation?

- I'm quite confused, the derivatives gotten above are depending on how long Sal drew his slope(50 votes)
- we don't care about how long is that line, we care about its slop (how steep or shallow is it) because for a line you can take any two points (now matter how far or how close) on it and you will have the same slop since it's a line.

and you can check it at2:09:

Sal took the points: (5,5) and (6,7)

so the slop is (7-5)/(6-5)=2/1=2

take another points for the same line for example:

(4,3) and (6,7)

so the slop (7-3)/(6-4)=4/2=2

which is THE SAME SLOP since it's THE SAME LINE.(12 votes)

- I do not get the concept and the definition of the derivative. What exactly is a derivative?(16 votes)
- Suppose you have a function f(x). Its derivative f'(x) describes the instantaneous rate of change of f(x) for any x in the domain. Suppose I told you that f(3)=7. Now you know where the function is at x=3, but you know nothing of its motion. Is it increasing? Decreasing? How quickly. If I tell you that f'(x)=10, that would indicate that at x=3, f(x) is increasing quickly. The intuition will become easier as you keep studying calculus. It might be worth checking out the videos on the relationship between limits and derivatives.(47 votes)

- Around5:26the answer to the second example is that g'(4) > g'(6). I guess it makes sense as g'(4) is less negative than g'(6)... but I find this confusing because the instantaneous rate of change at g'(6) is greater than at g'(4), and the example would make it seem like those instantaneous rates of change are what's being compared. Am I off base here?(23 votes)
- it's kind of similar to comparing velocity and speed(absolute value of velocity).

g'(6) has more "speed" than g'(4), but g'(4) and g'(6) are both negative, so g'(4) has a higher velocity than g'(6) because it's less negative.

i hope i haven't confused you.(18 votes)

- Why can't we differentiate acceleration again with respect to time?(6 votes)
- The derivative of acceleration with respect to time is a quantity called jerk, probably because the greater the absolute value of the rate of change in acceleration, the more "jerky" the motion is.(30 votes)

- On the first problem, the derivative of the constant turned out to be 2. But i have also heard that the derivative of a constant is always zero [so the d/dx (5) = 0]. Im really confused someone please help

Thanks(5 votes)- For the first problem, we're estimating the slope of f(x). If it were constant, the given graph would be a horizontal line. What might have thrown you off is that we're estimating the derivative at a single point. When people say that the derivative of a constant is zero, the "constant" is a function such that f(x)=c. Taking the derivative at a single point, which is done in the first problem, is a different matter entirely. In the video, we're looking at the slope/derivative of f(x) at x=5. If f(x) were horizontal, than the derivative would be zero. Since it isn't, that indicates that we have a nonzero derivative.(14 votes)

- Can someone explain where these tangent lines are coming from? What is their purpose?(5 votes)
- Tangent is a line which touches only one point on the curve. Knowing the slope of the tangent at one particular point gives us the instantaneous velocity at that point.

This video might help you.

https://www.khanacademy.org/math/ap-calculus-bc/derivative-introduction-bc/intro-to-diff-calculus-bc/v/newton-leibniz-and-usain-bolt(3 votes)

- At5:23, Sal says that g'(4) is > than g'(6) which does not make sense considering that g'(6) is much steeper than g'(4) and if you take the absolute values of both -1 and -3, you get |-1| < |-3|(It does not matter whether the slope is positive or negative. When it comes to the slope of a line, all that matters is its absolute value). So shouldn't the correct answer be g'(4) < g'(6)?.(0 votes)
- In math, a slope of a function is always considered from left to right, which gives us positive or negative slope. So it matters if the slope is negative or positive. It's true that their absolute values would have different effect but you cannot just do that. It's true the steeper the greater when you don't take negative and positive into consideration, but in math, we have to consider them as negative or positive.

g'(6) is much steeper comparing to g'(4), but there are both negative. Because of that g'(6) < g'(4)(9 votes)

- is there a way to figure out the derivative without just eyeballing it? like when the two derivatives are real close to compare(3 votes)
- Well there are derivate rules that you can use to get to the derivative. Getting the derivative by eyeballing the function only really works as long as you work with very simple functions.(4 votes)

- How does he know that Dy=2 and Dx=1 on the first example please someone explain me(2 votes)
- He just uses y step/x step. You use the formula to figure out the slope. Hope that explains things.(2 votes)

- how can you find the slope of a curve from a point in the middle of the parabola for example x=0 but the slope on the negative side is 0.5 but the right side is -0.5(2 votes)
- It depends on the curve. If the equation is centered around the origin, the slope at that point will be zero no matter what the vertical or horizontal stretch or shrink happens to be. As long as there is no horizontal shift, the "turning point" where the slope is 0 will be at x=0.(1 vote)

## 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.