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Math Help - Derivative of exponential function

  1. #1
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    Post Derivative of exponential function

    Hi, we've recently gone over how to get the derivative of exponential and logarithmic functions, except I couldn't make it to that class and we've already moved on. I have the problem as follows:

    If $10,000 is invested in a savings account offering 4% per year, compounded semiannually, how fast is the balance growing after 3 years?

    Things I know:
    The answer to this problem is $446.02 per year, in the back of the book.

    The compound interest formula f(t) = P(1 + r/m)^mt is used.
    I have to get the derivative of the formula, and then just plug in the values.

    Problem is, I'm unsure how to get the derivative of the formula. I've done all I know how and I'm not getting the answer in the back.

    When I tried to find the derivative this is what I came up with, please correct any errors I have made.

    f '(t) = (P(1 + r/m)^mt) * ln(mt)

    Any help much appreciated.
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  2. #2
    MHF Contributor
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    Your f'(t) is not correct.

    f(t) = P(1 + r/m)^mt

    What are variables here?
    Only f(t) and t.
    The P, r and m are all constants ---they don't change in the given f(t).

    So the function is in the form
    f(t) = P[(a)^(g(t))]

    d/dx a^u = ln(a) *a^u *du/dx --------------***

    Hence,
    f(t) = P(1 + r/m)^mt
    f'(t) = P[ln(1 +r/m)]*[(1 +r/m)^mt]*m -----(i)

    Given:
    P = $10,000
    r = 4% per annum = 0.04
    m = 2 ----semi-annualy
    t = 3 years

    Plug those into (i),
    f'(t) = ($10,000)[ln(1 +0.04/2)]*[(1 +0.04/2)^(2*3)]*2
    f'(t) = $446.02
    Last edited by ticbol; October 10th 2007 at 11:31 AM.
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  3. #3
    Super Member

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    Lexington, MA (USA)
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    Hello, leviathanwave!

    You missed this differentiation formula: . \frac{d}{dx}\left(a^u\right) \:=\:a^u\cdot\frac{du}{dx}\cdot\ln a

    In baby-talk, the derivative is:

    . . (the original function) (derivative of the exponent) (natural log of the base)


    $10,000 is invested in a savings account offering 4% per year, compounded semiannually.
    How fast is the balance growing after 3 years?

    Things I know:
    The answer to this problem is $446.02 per year, in the back of the book.

    The compound interest formula f(t) \:= \:P\left(1 + \frac{r}{m}\right)^mt is used.
    I have to get the derivative of the formula, and then just plug in the values. . . . . no
    We are given: . P = 10,000,\; r = 0.04,\;m = 2

    Why not plug them in now? . f(t) \:=\:10,000\left(1 + \frac{0.04}{2}\right)^{2t}
    . . And our function is: . f(t) \:=\:10,000(1.02)^{2t}

    Differentiate: . \frac{df}{dt}\;=\;10,000(1.02)^{2t}\cdot 2\cdot\ln(1.02)

    Now evaluate at t = 3.

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