Main content
Calculus 1
Unit 2: Lesson 9
Derivatives of cos(x), sin(x), 𝑒ˣ, 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 𝑒ˣ
- Derivative of ln(x)
- Derivatives of 𝑒ˣ and ln(x)
- Proof: The derivative of 𝑒ˣ is 𝑒ˣ
- Proof: the derivative of ln(x) is 1/x
© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice
Proof: The derivative of 𝑒ˣ is 𝑒ˣ
AP.CALC:
FUN‑3 (EU)
, FUN‑3.A (LO)
, FUN‑3.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?(26 votes)
Where can I find the proof of limit as n→infinity (1+1/n)^n =e and limit as n→0 (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 → ∞} (1 + 1/n)^n = lim_{1/n → 0} (1 + 1/n)^(n) = lim_{x → 0} (1 + x)^(1/x).(8 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 𝑓(𝑥) = ln 𝑥 and 𝑔(𝑥) = (1 + 1∕𝑥)^𝑥
we get 𝑓(𝑔(𝑥)) = ln(1 + 1∕𝑥)^𝑥
Because the natural log function is continuous, we have
lim[𝑥 → ∞] 𝑓(𝑔(𝑥)) = 𝑓(lim[𝑥 → ∞] 𝑔(𝑥))
= ln(lim[𝑥 → ∞] (1 + 1∕𝑥)^𝑥)(4 votes)
how/why is (1+1/n)^n equal to (1+n)^(1/n)? Is this just a basic law of exponents(2 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)
Technically, the function x^0-1 is its own derivative.(2 votes)
Any function of the form a·e^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->∞) (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
(𝑓(𝑥 + 𝛥𝑥) − 𝑓(𝑥))∕𝛥𝑥
is the slope of the line through the points
(𝑥, 𝑓(𝑥)) and (𝑥 + 𝛥𝑥, 𝑓(𝑥 + 𝛥𝑥))
Then, as 𝛥𝑥 → 0 the two points practically become one and the same, and our slope will be that of the tangent line at (𝑥, 𝑓(𝑥)).(4 votes)
- At4:01how did we take natural logs of both sides and only remained with delta x(1 vote)
There are multiple ways to answer this question. One way would be to say that raising a number to the power e and taking the natural logarithm of a number are inverse operations. Performing both operations simultaneously undoes both, so to speak.
Another way to answer is the exponent property of logarithms. The logarithm of (assume natural log, if not mentioned) a raised to the power b is equivalent to b multiplied by the logarithm of a.
log(a^b) = blog(a).
If we insert e in the place of a, using log(e) = 1, we get the same result.(2 votes)
7:23