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## APยฎ๏ธ/College Calculus AB

### Course: APยฎ๏ธ/College Calculus ABย >ย Unit 2

Lesson 8: Derivatives of cos(x), sin(x), ๐หฃ, and ln(x)- Derivatives of sin(x) and cos(x)
- Worked example: Derivatives of sin(x) and cos(x)
- Derivatives of sin(x) and cos(x)
- Proving the derivatives of sin(x) and cos(x)
- Derivative of ๐หฃ
- Derivative of ln(x)
- Derivatives of ๐หฃ and ln(x)
- Proof: The derivative of ๐หฃ is ๐หฃ
- Proof: the derivative of ln(x) is 1/x

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# Proof: The derivative of ๐หฃ is ๐หฃ

AP.CALC:

FUNโ3 (EU)

, FUNโ3.A (LO)

, FUNโ3.A.4 (EK)

e, start superscript, x, end superscript is the only function that is the derivative of itself!

(Well, actually, f, left parenthesis, x, right parenthesis, equals, 0 is also the derivative of itself, but it's not a very interesting function...)

The AP Calculus course doesn't require knowing the proof of this fact, but we believe that as long as a proof is accessible, there's always something to learn from it. In general, it's always good to require some kind of proof or justification for the theorems you learn.

## Want to join the conversation?

- At7:23, how did the limit got inside the logarithm function? It is getting hard for me to make sense for this step. It is like saying lim (x -> 0) cos(x) = cos (lim x->0 x).

How is that possible?

Can this thing be only applied to logarithm functions or is it generic for other functions also like cos, sin etc?(30 votes)- It's NOT a general rule, and I wish Sal spent some time explaining why it works in this
*particular case.*

โ โ โ

First of all, we're dealing with a*composite function.*

๐(๐ฅ) = 1โln ๐ฅ

๐(๐ฅ) = (1 + ๐ฅ)^(1โ๐ฅ)

โ

๐(๐(๐ฅ)) = 1โln((1 + ๐ฅ)^(1โ๐ฅ))

In general terms we are looking for

๐น = lim(๐ โ 0) ๐(๐(๐))

This means that we let ๐ approach zero, which makes ๐(๐) approach some limit ๐บ, which in turn makes ๐(๐(๐)) approach ๐น.

In other words:

๐บ = lim(๐ โ 0) ๐(๐)

๐น = lim(๐(๐) โ ๐บ) ๐(๐(๐)) = [let ๐ฅ = ๐(๐)] = lim(๐ฅ โ ๐บ) ๐(๐ฅ)

Now, if we use our definitions of ๐(๐ฅ) and ๐(๐ฅ), we get

๐บ = lim(๐ โ 0) (1 + ๐)^(1โ๐) = [by definition] = ๐

๐น = lim(๐ฅ โ ๐) 1โln ๐ฅ = [by direct substitution] = 1โln ๐ = 1

Note that ๐น was given to us by direct substitution, which means that in this*particular case*we have

lim(๐ฅ โ ๐บ) ๐(๐ฅ) = ๐(๐บ) = ๐(lim(๐ โ 0) ๐(๐))

โ โ โ

EDIT (10/28/21):

The reason this works is because lim ๐ฅโ0 ๐(๐ฅ) = ๐ (i.e. the limit exists)

and๐(๐ฅ) is continuous at ๐ฅ = ๐

According to the theorem for limits of composite functions we then have

lim ๐ฅโ0 ๐(๐(๐ฅ)) = ๐(lim ๐ฅโ0 ๐(๐ฅ))

Sal explains that theorem here:

https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-5a/v/limits-of-composite-functions(65 votes)

- How can e^x be the only function that is the derivative of itself? Doesn't f(x) = 19e^x also satisfy this property?(13 votes)
- When we say that the exponential function is the only derivative of itself we mean that in solving the differential equation f' = f. It's true that 19f = (19f)' but this isn't simplified; I can still pull the 19 out of the derivative and cancel both sides. You are correct in saying that the general solution is Ae^x where A is a real value; however, the "A" part isn't the main focus - the main focus is the exponential, since that's what varies and the constants don't.(16 votes)

- Where can I find the proof of limit as nโinfinity (1+1/n)^n =e and limit as nโ0 (1+n)^(1/n)=e?(7 votes)
- https://www.khanacademy.org/math/algebra2/exponential-and-logarithmic-functions/e-and-the-natural-logarithm/v/e-as-limit

or

https://mathcs.clarku.edu/~djoyce/ma122/elimit.pdf

The proof of the two formulas are the same:

lim_{n โ โ} (1 + 1/n)^n = lim_{1/n โ 0} (1 + 1/n)^(n) = lim_{x โ 0} (1 + x)^(1/x).(8 votes)

- how/why is (1+1/n)^n equal to (1+n)^(1/n)? Is this just a basic law of exponents(7 votes)
- Think about it like this:

it is completely legal for us to define one variable as some amount of another variable. Therefore, we can say that n=1/u, for example.

Let's say n=1/u

and

(lim n-> inf) e= (1+1/n)^n

Now let's rewrite this in terms of u. The limit will be that u gets very small and approaches 0, because this will cause the fraction 1/u to become very large. For n=1/u: if n approaches infinity, u must approach 0 for both sides to approach infinity.

(lim u-> 0) (1+u)^(1/u) (I simplified 1/(1/u) to just u)

This, therefore, is equivalent to the other definition of e, because all we have done is described the variable in a new way without adding in or taking away anything from the original equation, just looking at it differently.(6 votes)

- At7:23, is it that this is an application of the principle:

lim(x->a)[ f(g(x)) ] = f( lim(x->a)[g(x)] )

?(4 votes)- Yes, with ๐(๐ฅ) = ln ๐ฅ and ๐(๐ฅ) = (1 + 1โ๐ฅ)^๐ฅ

we get ๐(๐(๐ฅ)) = ln(1 + 1โ๐ฅ)^๐ฅ

Because the natural log function is continuous, we have

lim[๐ฅ โ โ] ๐(๐(๐ฅ)) = ๐(lim[๐ฅ โ โ] ๐(๐ฅ))

= ln(lim[๐ฅ โ โ] (1 + 1โ๐ฅ)^๐ฅ)(4 votes)

- Technically, the function x^0-1 is its own derivative.(2 votes)
- Any function of the form aยทe^x is its own derivative, and these are the
*only*functions that are their own derivatives. The zero function is just the special case where a=0.(8 votes)

- Hi - i am interested that sal says that e = (1+n)^1/n when I graphed y = (1+x)^1/x the graph converges to 1. What mistake have I made?(3 votes)
- What you may have missed is lim (n->0) for that definition. You are correct that lim (n->โ) (1+x)^1/x = 1, but lim (n->0) (1+x)^1/x = e.(3 votes)

- When/where do we learn that change of variables method?(3 votes)
- At3:35, Sal came up with n . Can the whole proof be shown without this n ? Why did he came up with this idea and not something else ?(3 votes)
- At1:42, how did he change the derivative into a limit?

How is that possible?

What is the formula of changing?(1 vote)- That is the definition of derivative as a limit.

The derivative at a point is the slope of the tangent line at that point.

You can verify for yourself that

(๐(๐ฅ + ๐ฅ๐ฅ) โ ๐(๐ฅ))โ๐ฅ๐ฅ

is the slope of the line through the points

(๐ฅ, ๐(๐ฅ)) and (๐ฅ + ๐ฅ๐ฅ, ๐(๐ฅ + ๐ฅ๐ฅ))

Then, as ๐ฅ๐ฅ โ 0 the two points practically become one and the same, and our slope will be that of the tangent line at (๐ฅ, ๐(๐ฅ)).(4 votes)