2^x derivitive

  1. I need to find the derivitive of y=2^x using the definition of derivitive.
     
  2. jcsd
  3. Hurkyl

    Hurkyl 16,089
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    What have you done so far? What does the definition of derivative say?
     
  4. HallsofIvy

    HallsofIvy 40,960
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    This was also posted in the calculus section and there are about 10 replies there.
     
  5. mathwonk

    mathwonk 9,849
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    I doubt this is a suitable problem for a novice. even showing convergence is tough. i will look at the other posted answers. there is a good reason people start from the integral definition of ln(x) to derive this result.
     
  6. HallsofIvy

    HallsofIvy 40,960
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    If f(x)= ax, the f(x+h)= aa+x= axah so
    f(a+ h)- f(a)= ax(ah- 1).

    The derivative is lim (f(x+h)- f(x))/h= axlim {(ah-1)/h}. Notice that that is ax time a limit that is independent of x. That is, as long as the derivative exists, it is ax times a constant. The problem is showing that the lim{(ah-1)/h} EXISTS! And then showing that, if a= 2, that limit is ln(2).

    Showing that that limit exists is sufficiently non-trivial that many people (myself included), as mathwonk said, prefer to define ln(x) as the integral, from 1 to x of (1/t)dt. From that, it is possible to prove all properties of ln(x) including (trivially) that the derivative is 1/x. Defining ex as the inverse function of ln(x) leads to all the properties of ex (including the fact that it is some number to a power!), in particular that its derivative is ex itself and, from that, that the derivative of ax is (ln a) ax.
     
  7. dunno if i'm missing the point here but...

    write
    y=2^x
    as
    y=exp(x.ln2)
    =>
    y'=ln2.exp(x.ln2)
     
  8. cepheid

    cepheid 5,190
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    No, you did exactly what HallsofIvy was advocating, he was just pointing out that the question asked for it to be solved using the definition of a derivative, which makes things much harder. Easier to approach things from the other way, starting by defining the integral of 1/x.
     
  9. "The derivative is lim (f(x+h)- f(x))/h= axlim {(ah-1)/h}. Notice that that is ax time a limit that is independent of x. That is, as long as the derivative exists, it is ax times a constant. The problem is showing that the lim{(ah-1)/h} EXISTS! And then showing that, if a= 2, that limit is ln(2)."

    tell me if i'm wrong, but it doesn't seems so hard to determine this limit..
    (a^h-1)/h = (exp (h*ln(a) )-1) / h
    = ( 1 + h*ln(a) + o(h*ln(a)) - 1 ) / h h->0
    = ln(a) + o(ln(a))
    so lim (a^h-1)/h = ln(a) .......
     
  10. HallsofIvy

    HallsofIvy 40,960
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    Yes, assuming that you know "(exp (h*ln(a) )-1) / h= ( 1 + h*ln(a) + o(h*ln(a)) - 1 ) / h " its easy to do it. Proving what you assumed is the hard part!
     
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