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Derivative of 2^x

  1. Feb 6, 2012 #1
    1. The problem statement, all variables and given/known data

    Let's take two problems, the derivative of

    2x

    and

    xπ

    For the second one the book says that answer is

    πxπ-1

    Well if you can do that for the above, then why not for the first problem?

    The book gives the answer to the first problem as

    2x ln 2

    why not x2x-1?

    essentially what i need to know for finding derivatives, there are two techniques.
    1. nx^n-1
    2. k^x ln k

    when do I use which technique?
     
  2. jcsd
  3. Feb 6, 2012 #2

    Curious3141

    User Avatar
    Homework Helper

    Because that rule only works when the power of the x term is a constant (independent of x).

    To differentiate 2x, express it as exln(2). This is of the form ekx, where k is a constant.
     
  4. Feb 6, 2012 #3

    Deveno

    User Avatar
    Science Advisor

    it makes a BIG difference whether x is "upstairs" (in the exponent), or "downstairs" (being exponentiated).

    if you go back to the definition:

    d(2x)/dx = limh→0 (2(x+h)-2x)/h

    you can see that you're not going to get an easy way to simplify.

    basically, e is the "natural base" for exponential functions, and other bases have logarithms as a "conversion factor":

    2x = (eln(2))x = eln(2)x

    which is of the form eax, so has derivative aeax, by the chain rule.
     
  5. Feb 6, 2012 #4

    Mark44

    Staff: Mentor

    2x and xn are very different types of functions. The first is an exponential function, so called because the variable is in the exponent. The second is a power function, so called because the variable is in the base, which is raised to a fixed power.

    There is no single differentiation rule, other than the definition, that covers both of these functions. You are misusing the power rule to conclude that d/dx(2x) = x2x-1. Each of the rules has "fine print" that says when it can be applied. Read the fine print.
     
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