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Math Help - Where am I going wrong?

  1. #1
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    Where am I going wrong?

    I'm not sure what I'm doing wrong here...

    S(t) = 20,000(1 + e^{-0.5t})


    Find the rate of change when t=1


    S'(t) = 20,000(1 + e^{-0.5t}) d/dx(-0.5t)
    = -10,000(1+e^{-0.5t})


    S'(1) = -10,000(1+ e^{-0.5[1]})
    = -10,000(1 + (0.606530659)
    = -10,000(1.606530659)
    = -16,065


    Apparently I should be getting -6,065 instead. I realize that this has something to do with the fact that the '1 + ...." is still in the expression, when it shouldn't be, but I'm not sure how I should be getting rid of it..
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  2. #2
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    Re: Where am I going wrong?

    According to which rules of differentiation do you think that (1+e^{f(t)})'=(1+e^{f(t)})f'(t)?
    Thanks from HallsofIvy
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  3. #3
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    Re: Where am I going wrong?

    Quote Originally Posted by emakarov View Post
    According to which rules of differentiation do you think that (1+e^{f(t)})'=(1+e^{f(t)})f'(t)?
    I'm going by the chain rule, according to my text-book. From what I understand by my textbook, if f(x) in e^f(x) is differentiable, then the derivative of e^f(x) = e^f(x) * dx f(x).

    I realize I'm going wrong here, and it's a bit of a pain because I'm doing this via distance education, which has been lacking at the best of times. I'd be stuffed without this forum.
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  4. #4
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    Re: Where am I going wrong?

    You are right about (e^{f(t)})' (up to some typos). What about \left(1+e^{f(t)}\right)'? Do you know that (g(t) + h(t))' = g'(t) + h'(t) and c' = 0 for a constant c?
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  5. #5
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    Re: Where am I going wrong?

    Quote Originally Posted by emakarov View Post
    You are right about (e^{f(t)})' (up to some typos). What about \left(1+e^{f(t)}\right)'? Do you know that (g(t) + h(t))' = g'(t) + h'(t) and c' = 0 for a constant c?
    Hmm, I was aware of it, I Remember that rule from when I was doing derivatives without an exponential. I didn't think of it during this question however.

    So, I'm guessing it should be something more like this..?

    S'(t) = 20,000(1 + e^{-0.5t}) * dx (0.5t)<br /> <br />
= -10,000(e^{-.5t})

    Due to S'(t) removing the constant value from the equation?
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  6. #6
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    Re: Where am I going wrong?

    What I can suggest is rewriting the derivative one step at a time. At each step do a single transformation and be sure to identify which of the rules given here you are using.

    S'(t) = 20,000(1 + e^{-0.5t}) * dx (0.5t)
    I still don't understand how you obtained the right-hand side. Also, the notation dx (0.5t) is incorrect. Even dt(0.5t) does not mean the derivative of 0.5t.
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  7. #7
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    Re: Where am I going wrong?

    Quote Originally Posted by emakarov View Post
    What I can suggest is rewriting the derivative one step at a time. At each step do a single transformation and be sure to identify which of the rules given here you are using.

    I still don't understand how you obtained the right-hand side. Also, the notation dx (0.5t) is incorrect. Even dt(0.5t) does not mean the derivative of 0.5t.
    I obtained the right side because I thought that S'(t) = S(t) * f '(x).

    If f(x) = -0.5t, then f '(x) = -0.5 ?
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  8. #8
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    Re: Where am I going wrong?

    Quote Originally Posted by astuart View Post
    I obtained the right side because I thought that S'(t) = S(t) * f '(x).
    First, it should be f(t), not f(x). I assume that f(t) denotes the exponent. Second, S'(t) = S(t) * f '(t) is false because it is not an instance of the chain rule. The correct instance would be R(f(t))' = R'(f(t)) f'(t) where R(x) = 20,000(1 + e^x). Also, it may be easier to factor 20,000 through first.
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