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Math Help - Let n be a natural number. Make a conjecture about the nth derivative of the function

  1. #1
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    Let n be a natural number. Make a conjecture about the nth derivative of the function

    Let n be a natural number. Make a conjecture about the nth derivative of the function f(x) = e^(ax). That is, what is the nth derivative of e^(ax).

    Then use mathematical induction to prove your conjecture.

    My conjecture is as follows: For every natural number a, the nth derivative of the function f(x) = e^ax is f^n (x) = a^n (e^(ax)).

    I have proved the initial step P(1):
    f'(x)=ae^(ax) = a^1 e^(ax) so that ae^(ax) = ae^(ax). This proves the initial step. Now here is where I am having trouble proving the inductive step, that is P(k+1) is true. Here is what I have so far

    ae^(ax) = a^(n+1) e^(ax)
    ae^(ax) = a*a^n*e^ax

    I used the fact that a^(n+1) = a*a^n... but I don't know where to go from here, or how to prove the inductive step... any help please?
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  2. #2
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    Re: Let n be a natural number. Make a conjecture about the nth derivative of the func

    we assume P(k).

    so f^{(k)}(x) = a^ke^{ax}.

    now we try to prove P(k+1). remember, we can only use P(k).

    f^{(k+1)}(x) = (f^{(k)}(x))^{\prime} = (a^ke^{ax })^{\prime} = (a^k)(e^{ax})^{\prime}, since a^k is just a constant, and we are assuming P(k).

    now use your "base case" to finish.
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  3. #3
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    Re: Let n be a natural number. Make a conjecture about the nth derivative of the func

    I'm not entirely sure of what the base case would be in this situation. Would I be right in saying the (k+1) case is: a*a^n*e^ax?
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    Re: Let n be a natural number. Make a conjecture about the nth derivative of the func

    the "base case" is simply P(1).

    P(k+1) would be: f^{(k+1)}(x) = a^{k+1}e^{ax}, which is what you're trying to prove.

    what you wrote is "almost" the P(k+1) case (except you used n instead of k), save for some algebraic manipulation.
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  5. #5
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    Re: Let n be a natural number. Make a conjecture about the nth derivative of the func

    I am stuck with the algebraic manipulation, as sad as that may sound. I have no idea where to go from: a*a^k*e^ax to show it is true for P(k+1) any help please?
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  6. #6
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    Re: Let n be a natural number. Make a conjecture about the nth derivative of the func

    what is (a)(a^k)? use the laws of exponents: (a^m)(a^n) = a^{m+n} for any a > 0, and all real m,n.
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