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## AP®︎/College Calculus AB

### Unit 2: Lesson 2

Defining the derivative of a function and using derivative notation- Formal definition of the derivative as a limit
- Formal and alternate form of the derivative
- Worked example: Derivative as a limit
- Worked example: Derivative from limit expression
- Derivative as a limit
- The derivative of x² at x=3 using the formal definition
- The derivative of x² at any point using the formal definition
- Finding tangent line equations using the formal definition of a limit

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# Formal and alternate form of the derivative

AP.CALC:

CHA‑2 (EU)

, CHA‑2.B (LO)

, CHA‑2.B.2 (EK)

, CHA‑2.B.3 (EK)

, CHA‑2.B.4 (EK)

Sal introduces two ways of writing the limit expression for the derivative of a function at a point. Created by Sal Khan.

## Want to join the conversation?

- Just a philosophical digression.

If mathematicians until today don't know how to define a process like division by zero, isn't it impossible to assume that if a division by zero has any kind of definition, humanity can never surely estimate.

It seems kind of strange that the financial sector, higher analysis and modelling relies on mathematical concepts which aren't rigourously defined.

That would mean that at any given point you can estimate that a function has some

supposed behaviour but since you can never instantaneously easily access that point parametrically you are after all only speculating.(74 votes)- You raised an interesting topic for discussion, and to answer that, I have to get back to one of your sentences.

"It seems kind of strange that the financial sector, higher analysis and modelling relies on mathematical concepts which aren't rigourously defined."

You might get the wrong impression about division by zero. Mathematicians weren't just saying "Okay, let's define a new operation and call it division- Oh wait, it doesn't work on zero- let's say it's undefined, because any sensible answer will break the rules of math". I admit it, it might have happened in the beginning- but in the last few years, mathematicians have been making extremely careful and formal definitions of math objects, including the division operation.

Division is built as a function (of two variables), that says: given a and b, return a/b. It was built in a way that doesn't allow 0 to be the denominator, so when we say that 1/0 is not defined, we mean that division is not built to fit in zero in there.

I'll give you an example- When I'll ask you what x to the power of a duck is, you'll think I'm crazy. Why? because exponention is not defined when talking about ducks. You say, "exponention is an operation that you put on numbers, not on animals". With the same arguement, I can tell you "division is an operation that you put on non-zero numbers as the denominator".

Not all mathematical operation need to be defined on all numbers, you just decide what numbers do you want the operation to work with.(270 votes)

- I understand both forms, but what is the point of the alternative form of derivative? What can the alternative form do, which the standard form can't?(23 votes)
- The alternate form can give you the numerical value of the derivative at a particular point (where x=a), rather than a general formula for the derivative. There are times when you might want to compute it that way (especially with more complex problems).(20 votes)

- What does derivative mean?(16 votes)
- The derivative of a function is the measure of change in that function. Consider the parabola y=x^2.

For negative x-values, on the left of the y-axis, the parabola is decreasing (falling down towards y=0), while for positive x-values, on the right of the y-axis, the parabola is increasing (shooting up from y=0).

If you find the derivative of y=x^2 (it will be explained in later videos but the derivative of x^2 is 2x) and substitute in a negative x-value, you will get a negative number, meaning you have negative change i.e. the function is decreasing. On the other hand, if you substitute in a positive x-value, you will get a positive number, meaning you have positive change i.e. the function is increasing.(20 votes)

- Can´t I make something like f(x+h) -f(x)/(x+h)-x -> f(x) + f(h) - f(x)/h -> f(h)/h ?(5 votes)
- f(x+h) IS NOT equal to f(x) + f(h). So, no, you cannot evaluate it that way.

Example:

f(x) = x²

f(x+h) = (x+h)² = x² + 2hx + h²

while f(x)+f(h) = x² + h²

Note: there are a few types of functions where f(x+h) happens to equal f(x) + f(h), but this is the exception to the rule.(18 votes)

- As far as I know,
`f'(a)`

is the slope of a tangent line at`x = a`

. Does anyone have any idea how we can apply this definition in real life? In your example, please tell me what is**slope**and what is**tangent line**?(7 votes)- Slope can be thought as "How far you go out horizontally and how far you go up vertically to get back to the line". Another way of saying that is "Change in y over change in x." A tangent line is a line that intersects the graph at exactly one point. Slope is often seen as a rate of change, so differentials and derivatives are seen as a rate of change. Physics is interested often times how fast a particle is moving in which ever way, so you set up a coordinate system to find position, then if you differentiate position you obtain velocity. But let's say you want to know how fast your velocity is changing - that's acceleration. You can graph velocity versus time and if you create a tangent line to one point that would be your instantaneous acceleration or your acceleration at that instant.(4 votes)

- Hey, isn't a tangent line just a secant line with two points in the same place? Before I learned calculus here on Khan Academy, I taught myself to approach a tangent line with a secant line whose points get closer and closer and closer in the same way you approach an undefined point on a function with a limit.(3 votes)
- That's a good way to look at it, and it's actually the basis of the typical definitions of the derivative. Geometrically, the derivative is the limit of the secant slope as the two points of the secant line get closer to each other, just like you described. I think Sal has a video on that entitled "Tangent slope as the limit of secant slope," or something to that effect.(4 votes)

- When you draw the tangent line to point a at the beginning of the video in both graphs, would it be possible to find take another random point on that line and find the slope of the tangent line? Since the tangent line is straight, wouldnt that meant the slope is the same at every point? Thank you.(4 votes)
- Good question – if you could draw the tangent line "nearly perfectly" then that would work. There are, however, a number of drawbacks (so to speak) of this strategy. I encourage you to try your proposed method on a few equations at a couple of different points and compare the answers you get with the actual answers.

Some shortcomings to a graphical strategy:

1) Tedious.

2) Error prone.

3) In situation where you need an equation it would be impractical (as well as exceedingly inaccurate) to make estimates from graphs.

– For many applications of derivatives (e.g. optimization), we need an equation not just the derivative at a single point.

– Sometimes we even want the second derivative and so an equation for the derivative is doubly necessary.(2 votes)

- at4:19shouldnt x approach zero

then and only,'I think ' we might get closer to zero

PLEASE correct if i am wrong

I am talking about the slope at point 'a' on the second graph(2 votes)- Since x = a+h, when h approaches 0, x approaches a.(4 votes)

- at1:23Sal cancels the x's and the equation remaining is:

the limit (as x approaches 0) of ( f(x+h) - f(x) )/h

As x approaches 0, doesn't the whole equation approach infinity? If this is true, then why isn't the derivative of every function infinity?(2 votes)- The limit is when h approaches 0, not x. When h approaches 0, f(x+h) approaches f(x) and the numerator, f(x+h)-f(x), approaches 0. The numerator and denominator both approach 0, so the limit is 0/0. 0/0 is one of the inderterminate answers, so it is not necessarily infinity. The answer depends on the function we put in the limit.(3 votes)

- At1:20when I'm finding the derivative of the secant line do I use limits?(Sorry if I worded it weird.)(2 votes)
- The derivative at a point is equal to the limit of the slope of the secant line between that point and a nearby point, as the nearby point approaches that point.(3 votes)

## Video transcript

So let's think
about how we could find the slope of
the tangent line to this curve right
over here, so what I have drawn in red, at
the point x equals a. And we've already seen
this with the definition of the derivative. We could try to find a
general function that gives us the slope of the tangent
line at any point. So let's say we have
some arbitrary point. Let me define some arbitrary
point x right over here. Then this would be the
point x comma f of x. And then we could
take some x plus h. So let's say that this
right over here is the point x plus h. And so this point would be
x plus h, f of x plus h. We can find the slope
of the secant line that goes between these two points. So that would be your
change in your vertical, which would be f
of x plus h minus f of x, over the change
in the horizontal, which would be x plus h minus x. And these two x's cancel. So this would be the
slope of this secant line. And then if we want to find the
slope of the tangent line at x, we would just take the
limit of this expression as h approaches 0. As h approaches 0, this
point moves towards x. And that slope of the secant
line between these two is going to approximate the
slope of the tangent line at x. And so this right over
here, this we would say is equal to f prime of x. This is still a function of x. You give me an arbitrary x
where the derivative is defined. I'm going to plug it into this,
whatever this ends up being. It might be some nice,
clean algebraic expression. Then I'm going to
give you a number. So for example, if
you wanted to find-- you could calculate
this somehow. Or you could even
leave it in this form. And then if you
wanted f prime of a, you would just substitute a
into your function definition. And you would say,
well, that's going to be the limit as h approaches
0 of-- every place you see an x, replace it with
an a. f of-- I'll stay in this color for now--
blank plus h minus f of blank, all of that over h. And I left those blanks so
I could write the a in red. Notice, every place where I
had an x before, it's now an a. So this is the derivative
evaluated at a. So this is one way to find
the slope of the tangent line when x equals a. Another way-- and
this is often used as the alternate form
of the derivative-- would be to do it directly. So this is the point
a comma f of a. Let's just take another
arbitrary point someplace. So let's say this
is the value x. This point right over here on
the function would be x comma f of x. And so what's the slope of the
secant line between these two points? Well, it would be change
in the vertical, which would be f of x minus f of a,
over change in the horizontal, over x minus a. Actually, let me do that
in that purple color. Over x minus a. Now, how could we get a better
and better approximation for the slope of the
tangent line here? Well, we could take the
limit as x approaches a. As x gets closer and
closer and closer to a, the secant line slope is
going to better and better and better approximate the
slope of the tangent line, this tangent line that
I have in red here. So we would want to take the
limit as x approaches a here. Either way, we're doing
the exact same thing. We have an expression for
the slope of a secant line. And then we're bringing those
x values of those points closer and closer together. So the slopes of those secant
lines better and better and better approximate that
slope of the tangent line. And at the limit, it does become
the slope of the tangent line. That is the definition
of the derivative. So this is the more standard
definition of a derivative. It would give you your
derivative as a function of x. And then you can then input
your particular value of x. Or you could use the alternate
form of the derivative. If you know that,
hey, look, I'm just looking to find the
derivative exactly at a. I don't need a
general function of f. Then you could do this. But they're doing
the same thing.