Main content

### Course: Differential Calculus > Unit 2

Lesson 3: Derivative definition- 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
- Limit expression for the derivative of function (graphical)

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Worked example: Derivative from limit expression

We explore a limit expression and discover that it represents the derivative of the function f(x) = x³ at the point x = 5. By analyzing the alternate form of the derivative, we gain a deeper understanding of tangent lines and their slopes. Created by Sal Khan.

## Want to join the conversation?

- It would be nice if the video focused more on finding the derivative at an indicated point or a separate video for it. I'm great at the Formal form but the alternate form is more difficult for me. Even just a simple f(x)= 1/x, a=2 explanation would be great. If someone could help me out that would be great as well.(20 votes)
- If I understand you're question correctly, you'd like to have an explanation of how to use alternate form to get the derivative at a point. Hopefully my explanation below isn't too long >n<. I also tried to explain how formal and alternate form are different ways of saying the same thing.

Let's do this example you have provided, f(x)=1/x, with a=2.

For the sake of demonstration, let's do the formal form first. Formal form is saying "I'm going to find the derivative by decreasing a distance from my point at 2 down to 0".

You would probably write it out like this:

f'(x)=lim (h->0) ((1/(2+h)-1/2)/(h)

I'll let you crack the algebra to save space, but unless I made an error you should eventually boil it down to this:

-1/2(2+h)

and as h approaches 0, you get -1/4.

Ok, now let's talk alternate form. Instead of taking a point h distance away from 2 and reducing that distance to 0, I'm going to say it another way- "I'm going to just take a point x, and bring it closer to 2." If you think about it, it's the same thing. Once x approaches 2, the distance between the two will be 0-just like with the distance h from 2 in formal form. It's just an alternate way of finding the same thing. To put it more succinctly, we're substituting h with x, and just saying that the closer we get x to 2 is the same thing as saying we're moving a point 2+h closer to 2 by reducing h to 0.

So let's write it out:

f'(x)=lim (x->a) (1/x-1/2)/(x-2)

If you do the algebra (hint: Get the denominators at the top to play nice first), you'll eventually get:

-1/2x

Plug in 2 for x (remember, we're having x approach 2), and you'll get the same thing as formal form- -1/4.

Hope that was what you were looking for and it wasn't too long. Let me know if this helps.(38 votes)

- Any tip
**to understand**the difference(s) between the formal form and the alternate one clearer and**to apply**them better? In what case(s) we should use the prior, and in what case(s) the latter is better?(15 votes)- Generally, we will use the formal form of a derivative when trying to find elementary derivatives like finding the derivative of y = x, or the derivative of y = x^2. The alternate form is still useful, although it is primarily used in finding slopes of lines, not curves.

Differences between formal form and alternate form are in how the limits are defined. In both, we are letting two points approach each other, but one uses lim h --> 0 and the other uses lim x --> a.(15 votes)

- I calculated the slope of the tangent line of f(x)=x^3 at x=5 to be 75. Can anyone confirm this for me?(7 votes)
- That is correct. The derivative of x^3 is 3x^2, and if you plug in x=5, that comes out to 75.(8 votes)

- The secant line is found by (Y2-Y1 / X2-X1) I always thought the (X2 , Y2) were the point furthest to the right? But in this video (4:52) he is coming from the left. How do we know what values to put in ( f(x)-f(a) / x-a ) ? or does it matter what order they are in?(5 votes)
- The secant line is basically the slope of the 2 points. In other words, changes in Y over changes in X. It doesn't matter from which direction you approach it from as long as you are consistent with your order pair, meaning, let (x,f(x)) and (a,f(a)) be the two points, you can use f(x)-f(a)/x-a or f(a)-f(x)/a-x.

Generally, in the book it will teach you that f(x) is on the order pair from the right and > f(a) this is because it's more intuitive when doing the subtraction. But just remember that the secant line is just the slope between any two points on the graph. And if you know how to find the slope of any two points, then you can find the secant line.(5 votes)

- what is a prime function with the symbol ' such as f'(a) ?(3 votes)
- f'(a) denotes the first derivative of f(a)(9 votes)

- Knowing only the formal and the alternative form of the derivative, are we supposed to know how to evaluate f'(x) or f'(a)? Specifically, I am having troubles simplifying (x^3 - 125)/(x-5) so that I can take the limit of it as x approaches 5. If so, I might have just missed something and I will go back some videos to revise.(4 votes)
- If you took the derivative at that point would you get a slope of 75?(4 votes)
- Yup! Since when
`f(x) = x^3`

:`f'(x) = 3x^2`

`f'(5) = 3(5)^2 = 3*25 = 75`

(3 votes)

- Why does x to the 3rd equal 5 to the 3rd(3 votes)
- It's because we're taking the limit as x approaches 5. As the function is a polynomial, we can directly take x = 5 as polynomials are always continuous.(2 votes)

- How do you find f(x)= 14x+13, if f'(1) or f prime of 1(1 vote)
- I'm not sure I completely understand your question, but since your function is linear, the slope is constant, so the derivative is also constant. Your equation is of the linear form: y=mx+b where m is the slope. In your case it is 14, so the derivative is f'(x)=14.

f'(1)=14 too(6 votes)

- I thought the derivative was going to be an actual value. Why was it kept in that form?(2 votes)
- Normally if a function is differentiable, then you'd want to write down what that resulting function is so that you can compute it at any point you'd like - instead of a numerical value. For instance if f(x) = x^2 then f'(x) = 2x. This is true for any value of x.(3 votes)

## Video transcript

The alternate form of the
derivative of the function f, at a number a,
denoted by f prime of a, is given by this stuff. Now this might look a
little strange to you, but if you really think
about what it's saying, it's really just
taking the slope of the tangent line
between a comma f of a. So let's imagine some
arbitrary function like this. Let's say that that
is-- well I'll just write that's our function f. And so you could
have the point when x is equal to a-- this is our
x-axis-- when x is equal to a, this is the point a, f of a. You notice a, f of a. And then we could take the slope
between that and some arbitrary point, let's call that x. So this is the point x, f of x. And notice, the numerator
right here, this is just our change in the
value of our function. Or you could view that as the
change in the vertical axis. So that would give you this
distance right over here. That's what we're doing
up here in the numerator. And then in the
denominator, we're finding the change in
our horizontal values, horizontal coordinates. Let me do that in
a different color. So the change in the horizontal,
that's this right over here. And then they're trying to find
the limit as x approaches a. So as x gets closer and
closer and closer and closer to a, what's going to happen
is, is that when x is out here, we have this secant line. We're finding the slope
of this secant line. But as x gets closer and
closer, the secant lines better and better and
better approximate the slope of the tangent line. Where the limit, as x approaches
a, but doesn't quite equal a, is going to be--
this is actually our definition of
our derivative. Or I guess the alternate form
of the derivative definition. And this would be the slope of
the tangent line, if it exists. So with that all
that out the way, let's try to answer
their question. With the Alternative Form
of the Derivative as an aid, make sense of the
following limit expression by identifying the function
f and the number a. So right here, they want to find
the slope of the tangent line at 5. Here they wanted to find the
slope of the tangent line at a. So it's pretty clear
that a is equal to 5. And that f of a is equal to 125. Now what about f of x? Well here, it's a limit
of f of x minus f of a. Well here it's the limit as
x to the third minus 125. And this makes sense. If f of x is equal
to x to the third, then it makes sense that f of 5
is going to be 5 to the third, is going to be 125. And we're also taking up here
the limit as x approaches a. Here we're taking the
limit as x approaches 5. So this is the derivative
of the function f of x is equal
to x to the third. Let me write that down
in the green color. x to the third at the
number a is equal to 5. And so we can imagine this. Let's try to actually graph it,
just so that we can imagine it. Actually, I'll do
it out here, where I have a little bit better
contrast with the colors. So let's say that is my y-axis. Let's say that
this is my x-axis. I'm not going to quite
draw it to scale. Let's say this right
over here is the 125. Or y, this is when y equals 125. This is when x is equal
to 5, so they're clearly not at the same scale. But the function is going
to look something like this. We know what x to
the third looks like, it looks something like this. So here, our a is equal to 5. This point right
over here is 5, 125. And then we're taking the
slope between that point and an arbitrary x-value. Or I should say an arbitrary
other point on the curve. So this right over here
would be the point, we could call that
x, x to the third. We know that f of x is
equal to x to the third. And let me make it clear. This is a graph of y is
equal to x to the third. And so this expression, right
over here, all of this, this is the slope between
these two points. And as we take the
limit as x approaches 5, so right now this
is our x, as x gets closer and closer
and closer to 5, the secant lines are
going to better and better approximate the slope of the
tangent line at x equals 5. So the slope of a tangent line
would look something like that.