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Very simple exponential function quesiton

  1. Aug 26, 2012 #1
    The derivative of e^(2x):

    let y = e^(2x), let u = 2x, so y = e^u

    chain rule: du/dx * dy/du = 2*e^u = 2e^u = 2e^(2x)

    this is the solution copied from my book, my question is why do they let u = 2x? is e^u the same as e^x? If so then wouldn't all derivatives of the exponential functions be in the form of f`(x)e^(f(x))??
     
  2. jcsd
  3. Aug 26, 2012 #2

    phyzguy

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    Yes, the derivative of e^(f(x)) is f'(x) e^(f(x)), regardless of what f(x) is. Does that answer your question?
     
  4. Aug 26, 2012 #3
    Why in the solutions do they let u = 2x? Is differentiating e^u the same as differentiating e^x?
     
  5. Aug 26, 2012 #4

    vela

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    You have to keep in mind which variable you are differentiating with respect to.
    \begin{align*}
    \frac{d}{du} e^u &= e^u \\
    \frac{d}{dx} e^u &= e^u\frac{du}{dx}
    \end{align*}
     
  6. Aug 26, 2012 #5

    Ray Vickson

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    They are just using (d/dx) e^f(x) = f'(x) e^f(x) [an equation which YOU wrote!], with f(x) = 2x.

    RGV
     
  7. Aug 26, 2012 #6
    thanks for all the replies, how did you figure out the second line of your reply? (differentiating e^u with respect to x) as I don't think I've seen that before (I've only just started calc).
     
  8. Aug 26, 2012 #7

    vela

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    It's just the chain rule.
     
  9. Aug 26, 2012 #8
    My book says this about the chain rule: BtiOv.png

    How does what you said about differentiating e^u with respect to x apply to what my book said? Could you explain further as I'm confused.
     
  10. Aug 26, 2012 #9
    Because u is a function of x.

    So e^u's derivative, by the chain rule, is u'e^u.
     
  11. Aug 26, 2012 #10

    vela

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    Can you explain what's confusing you? I mean, we can keep telling you "It's the chain rule," but it's hard to see what's troubling you.
     
  12. Aug 26, 2012 #11
    I just don't really understand the chain rule.
     
  13. Aug 26, 2012 #12

    phyzguy

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    Perhaps it would help to write u as u(x). Then can you see that the chain rule gives:
    [tex]\frac{d}{dx}e^{u(x)} =e^{u(x)}\frac{du}{dx}[/tex]

    If u is not a function of x, then:
    [tex]\frac{d}{dx}e^{u} =0[/tex]

    Does this help?
     
  14. Aug 26, 2012 #13
    In my opinion, written out, it looks much more complicated than it is.

    Take the derivative of the outer function, still evaluated for the inner function, then multiply the whole thing by the derivative of the inner function.

    One thing that is cool about the derivative rules is that they always work. For example, take a function that you know the derivative of, like 2x. You can apply the chain rule, product rule, etc. to that and you will get the correct result.

    2(x).

    Derivative of the outer function, evaluated for x, multiplied by the derivative of x, is just 2(1), or 2.
     
  15. Aug 26, 2012 #14
    Not really, I don't understand if u is not a function of x then that makes the derivative = 0,

    Yes, but I don't really see how this applies when differentiating with respect to another variable as above

    thank you for everyone who has bared with me, I'll probably sleep as I just don't understand it at all.
     
  16. Aug 26, 2012 #15
    Put it this way:

    If I say u = 2x, then I ask you to find the derivatives of e^(2x) and e^(u), you had better get the same answer.

    If you apply the chain rule to e^(2x) and not to e^(u) just because u is a "single letter," you won't get the same result.

    What's the derivative of e^(6)?
     
    Last edited: Aug 26, 2012
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