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 as a limit

Discover how to apply the formal and alternate forms of the derivative in real-world scenarios. We'll explore the process of finding the slope of tangent lines using both methods and compare their effectiveness in solving calculus problems. Let's dive into the practical side of derivatives to deepen our understanding. Created by Sal Khan.

## Want to join the conversation?

- i guess im not understanding something here. Its seems in both the formal and alternate limits as h approaches 0 and x approaches e 0/0. So does the limit not exist or did i skip a step?(54 votes)
- The value (e * 0) / 0 doesn't exist, but the limit (ln(e + h) - ln(e)) / h as h approaches 0 does exist. That is why we use limits: they allow us to get a handle on values without breaking algebra.(107 votes)

- Why did someone make the alternate if there was a formal already?(17 votes)
- There are always many ways to solve Math Questions.

Therefore in the subtopic Derivatives there are also many ways to solve the same question.

The Formal way is used when you want a general equation in which you just put a x value and get the Derivative at that particular x.

This can be used when you have to find many Derivatives on the same Function.

But when you want the value at just one particular x value you can use Alternate method.

This will save time for that question.

Whichever you use, you'll get the same answer.

Sal told us the Alternate method so that we can clear our concepts.

Once our concepts are clear we can understand everything nicely. :)(40 votes)

- is this related to the fact that when u zoom into a curve to high levels, the curve appears to be a straight line, and then you can take points on the straight line (they will be very close to each other when zoomed out), and then find the slope?(20 votes)
- Very astute observation! This is exactly how derivatives work. This definition goes into even further (i.e. more rigorous) detail in real analysis.(23 votes)

- What is the derivative of f(x)=ln x?(9 votes)
- it is
`1/x`

. That might feel a bit odd.

See:

https://www.khanacademy.org/math/calculus/differential-calculus/der_common_functions/v/proof--d-dx-ln-x----1-x?v=yUpDRpkUhf4

or

https://www.khanacademy.org/math/calculus/differential-calculus/der_common_functions/v/proofs-of-derivatives-of-ln-x--and-e-x?v=3nQejB-XPoY

where Khan explains why this is.(17 votes)

- What;s the difference between a normal logarithm and a natural logarithm?(2 votes)
- Logarithm has different bases. Usually, we write them with the base. However, for log base of e (a constant similar to pi) is called natural logarithm, and written as ln without a base (and we know it has the base of e). Similarly, there is a case of log (without any base) is understood to be log base of 10. So natural log is just a normal log with a special base (constant e).(12 votes)

- which video tells me to use these formulas to actually find the derivative ?(7 votes)
- Why x approaches 0 in formal definition?(2 votes)
- We can conceptualize the tangent line as being a secant line where the two points get closer an closer. If we put the distances between the secant lines close enough (if the difference between them approaches 0) then we will have a tangent line at exactly one point on the function, which we define to be the derivative.(7 votes)

- how do you find the f'(x) of f(x)=x^e^x ?

this is a question given in an exercise book i have that i am really confused with. so i really hope for someone to explain it to me..(2 votes)- To make this easier to follow, I will replace f(x) with y, though you don't really need to do this.

y = x^e^x

ln y = ln(x^e^x)

ln y = (e^x)(ln x)

Taking the implicit derivative:

(1/y) dy = [ (e^x)(ln x) + (e^x)(1/x) ] dx

dy/dx = y [ (e^x)(ln x) + (e^x)(1/x) ]

dy/dx = ( x^e^x) [ (e^x)(ln x) + (e^x)(1/x) ](6 votes)

- For the alternate definition of the Derivative why can't you say the limit of e approaches x. Isn't the limit if x approaching e the same thing as the limit of e approaching x? Sal discusses at5:24(3 votes)
- e can't approach x, or do anything else, because it's a constant. We can't select different values for it.(3 votes)

- at5:42when the final derivative is Ln(x)-1/x-e as X approaches to e,MY question is that how will we solve it as a limit? I mean how can we solve it using algebraic methods like we did in the Limits topic?(3 votes)
- To prove this limit (calculate from first principles) you need to know the limit based definition of
`e`

... this is covered starting here:

https://www.khanacademy.org/math/ap-calculus-ab/ab-derivatives-advanced/ab-adv-derivatives-opt-vids/v/proof-of-derivative-of-ex

The next video goes through the actual proof (but you should do the previous video first):

https://www.khanacademy.org/math/ap-calculus-ab/ab-derivatives-advanced/ab-adv-derivatives-opt-vids/v/proof-of-derivative-of-ln-x(3 votes)

## Video transcript

Let's say that f of x is
equal to the natural log of x, and we want to figure out what
the slope of the tangent line to the curve f is when x
is equal to the number e. So here, x is equal
to the number e. The point e comma
1 is on the curve. f of e is 1. The natural log of e is 1. And I've drawn the slope
of the tangent line, or I've drawn the tangent line. And we need to figure
out what the slope of it is, or at least come up
with an expression for it. And I'm going to come up
with an expression using both the formal definition
and the alternate definition. That will allow us to
compare them a little bit. So let's think about first
the formal definition. So the formal
definition wants us to find an expression for the
derivative of our function at any x. So let's say that this is some
arbitrary x right over here. This would be the
point x comma f of x. And let's say that this is--
let's call this x plus h. So this distance right
over here is going to be h. This right over
here is going to be the point, x plus
h f of x plus h. Now, the whole underlying
idea of the formal definition of limits is to find the
slope of the secant line between these two
points, and then take the limit as
h approaches 0. As h gets closer and
closer, this blue point is going to get closer and
closer and closer to x. And this point is going to
approach it on the curve. And the secant line
is going to become a better and better and
better approximation of the tangent line at x. So let's actually do that. So what's the slope
of the secant line? Well, it's the change in
your vertical axis, which is going to be f of x
plus h minus f of x-- over the change in
your horizontal axis. And that's x plus h minus x. And we see here the
difference is just h. Over h. And we're going to
take the limit of that as h approaches 0. So in the case when f of
x is the natural log of x, this will reduce to the
limit as h approaches 0. f of x plus h is
the natural log of x plus h minus the natural log
of x, all of that over h. So this right over here,
for our particular f of x, this is equal to f prime of x. So if we wanted to evaluate
this when x is equal to e, then everywhere we
see an x we just have to replace it with an e. This is essentially
expressing our derivative as a function of x. It's kind of a
crazy-looking function of x. You have a limit
here and all of that. But every place you see an x,
like any function definition, you can replace
it now with an e. So we can-- let me just do that. Whoops. I lost my screen. Here we go. So we could write
f prime of e is equal to the limit as h
approaches 0 of natural log-- let me do it in the
same color so we can keep track of things--
natural log of e plus h-- I'll just leave
that blank for now-- minus the natural log of
e, all of that over h. So just like that. This right over here, if
we evaluate this limit-- if we're able to and
we actually can-- if we are able to
evaluate this limit, this would give us the slope
of the tangent line when x equals e. This is doing the
formal definition. Now let's do the
alternate definition. The alternate
definition-- if you don't want to find a
general derivative expressed as a function of x
like this and you just want to find the slope
at a particular point, the alternate definition kind of
just gets straight to the point there. So what they say is
hey, look, let's imagine some other x value here. So let's imagine
some other x value. This right over here is
the point x comma-- well, we could say f of x or we could
even say the natural log of x. What is the slope of the secant
line between those two points? Well, it's going to be
your change in y values. So it's going to be
natural log of x minus 1-- let me do that red color--
over your change in x values. That's x minus e. So that's the slope of the
secant line between those two points. Well, what if you want
to get the tangent line? Well, let's just take the
limit as x approaches e. As x gets closer and
closer and closer, these points are going to get
closer and closer and closer, and the secant line is
going to better approximate the tangent line. So we're just going to take
the limit as x approaches e. So either one of this. This is using the formal
definition of a limit. Let me make it clear that that
h does not belong part of it. So we could either do it
using the formal definition or the alternate definition
of the derivative.