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Derivative of 𝑒ˣ

AP.CALC:
FUN‑3 (EU)
,
FUN‑3.A (LO)
,
FUN‑3.A.4 (EK)
The derivative of 𝑒ˣ is... well... 𝑒ˣ. This is a very special property lies at the heart of our work with exponential functions.

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  • starky sapling style avatar for user 20leunge
    How would you use the "lim h-->0 f(x+h) - f(x)/h" strategy to solve for the derivative of e^x? I got 0/0, but it doesn't look like factoring or using conjugates work.
    (7 votes)
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    • blobby green style avatar for user Tom Ahahaha
      Factoring will work!

      f(x)=e^x : this will be our original equation that we want to differentiate to achieve the general formula. As noted by this video, the general formula for this equation is the equation itself: e^x. Let's prove it using the general limit notation!

      First, plug in (x) and (x+h) into the exponent.
      f(x)= e^x
      f(x+h)=e^(x+h)

      Now that we have established the terms for the limit notation, we can continue:
      lim h-->0 (e^(x+h) - e^x)/h
      lim h-->0 (e^x times e^h - e^x)/h : bare with me, I know it's difficult to illustrate this but remind yourself that a^(b+c) is equivalent to a^b times a^c, as long as the base is the same, according the the exponential laws. Thus, e^(x+h) can be separated as e^x times e^h, since multiplying the two bases with the same exponent simply means to add the exponents together.

      lim h-->0 [e^x(e^h - 1)]/h : factor out e^x, since this is the common factor. If you did not recognize this operation, you can set e^x = a
      *** lim h-->0 (a times e^h - a)/h
      *** lim h--->0 [a (e^h - 1)]/h : where e^x = a

      lim h-->0 [e^x(e^h - 1)]/h : the coefficient, e^x does not depend on the denominator; it can be interpreted as (e^x)/1 times (e^h - 1)/h. Thus, move the e^x in front of the lim.

      e^x times lim h-->0 (e^h - 1)/h : approximate the value of the limit by picking a number very close to zero to input into 'h' value. Let's choose .0001 (you can choose .01, .001, .0000000000000000000001...but you will find that as h gets closer to zero, the value of the limit gets more accurate).

      e^x times lim h-->0 (e^0.0001 - 1)/0.0001 : the value of the limit is 1
      e^x times 1
      f'(x)= e^ x : this proves that the derivative (general slope formula) of f(x)= e^x is e^x, which is the function itself. In other words, for every point on the graph of f(x)=e^x, the slope of the tangent is equal to the y-value of tangent point. So if y= 2, slope will be 2. if y= 2.12345, slope will be 2.12345
      (22 votes)
  • leafers ultimate style avatar for user William Wu
    Wouldn't e^x fall under the power rule? So why does it behave differently?
    (5 votes)
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  • stelly yellow style avatar for user lemonomadic
    But shouldn't be the derivative of e^x = xe^(x-1) using Power Rule?
    (2 votes)
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  • blobby green style avatar for user Anwesha Donna
    Does that mean, at y = infinity, the graph will be a vertical line and thus, the slope of e^x at y = infinity will be undefined?
    (4 votes)
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  • purple pi purple style avatar for user The first integral proponent
    How did he got the slope so fast??
    (1 vote)
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  • blobby green style avatar for user SDJ7448
    Is this function, e^(x) really differentiable?if yes then how. My math teacher says it isn't actually differentiable.
    (0 votes)
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  • male robot hal style avatar for user tennisavatar
    What would the derivative of e^(-5x) be then? Would it be the same thing?
    (1 vote)
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  • blobby green style avatar for user softball.summers3
    Is this true as well for f(x)= 2^x? I dont know how to find the derivative of this. My professor wants us to use the definition of a derivative to solve.
    (2 votes)
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  • blobby green style avatar for user anuvanadkar19
    How would you differentiate e^-x?
    (1 vote)
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  • piceratops seed style avatar for user nizodahab
    Doesn't this apply to any function where x is the exponent, because using the power rule, if you multiply x by the constant, and raise the constant to x-1, then it's just that constant to the power of x again.
    (1 vote)
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Video transcript

- [Instructor] What we have right over here is the graph of Y is equal to E to the X and what we're going to know by the end of this video is one of the most fascinating ideas in calculus and once again it reinforces the idea that E is really this somewhat magical number. So we're gonna do a little bit of an exploration. Let's just pick some points on this curve of Y is equal to E to the X and think about what the slope of the tangent line is or what the derivative looks like and so let's say when Y is equal to one or when E to the X is equal to one, this is the case when X is equal to zero. Well, the slope of the tangent line looks like it is one, which is curious because that's exactly the value of the function at that point. What about when E to the X is equal to two right over here? Well here, let me do it in another color, the slope of the tangent line sure looks pretty close, sure looks pretty close to two. What about when E to the X is equal to 1/2? So that's happening right about here. Well, it sure looks like the slope of the tangent line is about 1/2. We could try what happens when E to the X is equal to five? Well, the slope of the tangent line here sure does look pretty close, sure does look pretty close to five and so just eyeballing it, is it the case that the slope of the tangent line of E to the X is the same thing, is E to the X? And I will tell you and this is an amazing thing that that is indeed true, that if I have some function, F of X, that is equal to E to the X and if I were to take the derivative of this, this is going to be equal to E to the X as well or another way of saying it, the derivative with respect to X of E to the X is equal to E to the X and that is an amazing thing. In previous lessons or courses, you've learned about ways to define E and this could be a new one. E is the number that where if you take that number to the power of X, if you define a function or expression as E to the X, it's that number where if you take the derivative of that it's still going to be E to the X. And what you're looking here, this curve, it's a curve where the value that's Y value at any point is the same as the slope of the tangent line. If that doesn't strike you as mysterious and magical and amazing just yet, it will. Maybe tonight you'll wake up in the middle of the night and you'll realize just what's going on. Now, some of you might be saying okay, this is cool, you're telling me this, but how do I know it's true? In another video, we will do the proof.