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Proof that the derivative of e^x is e^x.
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Differential calculus on Khan Academy: Limit introduction, squeeze theorem, and epsilon-delta definition of limits.
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2:57 Almost all explanations of the derivative of f(x)-e^x skip through the chain rule and confuse students.
The chain rule is for composite functions; a function f of some other function g(x). f(g(x))= f of g of x. Nested functions.
Such as ln(u)= outer function f(u)=ln(u). Let u=e^x=g(x)=the inner function, in this case.
d/dx(f(g(x))=f'(u) · g'(x) = d/d(u)(f(u)) · d/dx(g(x))
d/du(ln(u))= 1/u=1/e^x , because u=e^x. d/dx(g(x))=d/dx(e^x)=?
Multiply those two answers: d/dx( ln(e^x) = (1/e^x) · (d/dx(e^x)) = chain rule answer.
From the red calculation which says, ln(e^x)=x, inverse functions, therefore d/dx( ln(e^x))=d/dx(x)=1.
That means the chain rule answer (1/e^x) · (d/dx(e^x)) = 1 also.
Multply both sides by e^x, then d/dx(e^x)=1· (e^x), d/dx(e^x)=e^x.
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15years 424285 125 1406likes
If we assume that d/dx ln(x) = 1/x then let y = e^x => x = ln(y) => 1 = y'/y (differentiating both side wrt x) => y' = y = e^x.
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Wait the derivative of lnx is the (derivative of x)/x that's why it's 1/x
The point is the derivative of ln(f(x)) is (the derivative of f(x))/f(x)
So d(ln e^x)/dx is : (the derivative of e^x)/e^x
If m wrong hope someone correct me
Xne
Horrible explanation. Who is this guy?
You can use the definition of a derivative to solve this which would make the proof seem more valid.
sir find[d/dx(x^e)]
"I think the school or the nation should take a national holiday and respond to this."
The last step is wrong
U r multiplying e^x on both sides but on the RHS u have to multiply it out of the derivative that is d/dy. e^x.1/e^x.that step is wrong
#impressed
Seems like someone enjoyed quite a few drinks before recording this video.
Your title is misleading somehow… d(e^x)/dx = e^x 😉
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so the video used d/dx (e^x) in a chain rule to prove that d/dx(e^x) = e^x. I don't think that would be considered valid proof since ur using the result with chain rule to prove the result.
So I spent an hour proving to myself that the derivative of lnx=1/x.. Then I was gonna try the derivative of e^x and I got lazy.. Now I wish I did it alone because it looks so easy.. But I have just one question.. Isn't d/dx(e^x) it's own term ? In a proof, you can't assume the right answer in it, you must prove the right answer. If we don't know the derivative of e^x is e^x yet, why are you saying that (e^x(d/dx(e^x)))/e^x=e^x. I'm only in algebra II, so I could really be wrong here.. But does anyone else feel the same ? I'm a bit confused..
Derivative of e^x is the y coordinate
Derivative of 1/2*x^2 is the x coordinate
I think they should be equally celebrated
thank Youuuuuu
e^(iπ)+1=0
4:14 This is why I love Sal.
if d/dx lne(e^x)=1/e^x and d/dx e^x=1/e^x then
d/dx lne(e^x) = d/dx e^x
doesn't that mean that lne(e^x) = e^x
lne(e)*x=e^x
x=e^x
has anyone else noticed that Sal sometimes sounds drunk in his videos?
When I try and find the integral from 0-1 for this function, i run into the issue of receiving -infinity for 0 with the way i integrated it. So i figure im doing something wrong.
How do you solve for exponents such as negative square root of 'x' ( e^-(x^(1/2)) )
1/(e^x) is not being derived. It e^x is being derived, which means you're in the clear.
The poof goes like this:
1. Take the derivative of ln(e^x) by first using log rules to show ln(e^x) = x. d/dx (x) = 1.
2. Now that you know that, go through taking the derivative of ln(e^x) a second time. This time, you use the chain rule.
3. AFTER using chain rule, you get:
(1/e^x) * ( d/dx(e^x) ) = d/dx( ln(e^x) ) = 1 [From Step 1]
Or simply:
(1/e^x) * ( d/dx(e^x) ) = 1
Multiply both sides by e^x to get rid of the fraction, and you're done.
I understand the chain rule but how did you make the jump at 2:43
Also when you multiplied both sides by e^x, you can't because 1/(e^x) is being derived, or so I thought?
Two things I don't understand, I thought we were trying to prove d/dx(e^x), but in this video he starts by doing it as d/dx(ln(e^x)). I'm assuming ln(e^x) = e^x?
Next he's doing the chain rule to ln(e^x). First step is easy, taking the derivative of the inside which is e^x. Then the next step, which is supposed to be the exponent – 1 (n – 1), he gets 1/e^x instead. He got that from taking the derivative from lnx = 1/x… but I don't understand how that correlates.
I thought d/dx[e^x] = 42 🙂
Lol I'm stupid u guys are nerds
I guess this formula found by accident..
but because h cannot be 0 it cannot differentiate exactly to e^x, but i could be wrong :L :/
I'm pretty sure you went wrong somewhere here, you differentiated ln(e^x) to get 1, then differentiated it a different way to get 1/e^x, now assuming that makes sense you say that they must equal each other so 1/e^x=1 so 1=e^x you cant just scribble out the 1. a way to prove it is to say on the curve of e^x we can choose two points, p and p+h where h is the difference in the x axis between the two points so dy/dx((e^(p+h)-e^p)/h) as h gets closer to 0 it becomes effectively e^x
…derberdbebrerebdederebeberrrrebed.
E to the X!
Yes, well done. How is that relevant?
Correction: the result would be derivative of WHAT_EVER = WHAT_EVER