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Math Help - Derivatives of Exponential Functions

  1. #1
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    Derivatives of Exponential Functions

    Hi,

    I was wondering if someone could check my work for the following two short problems:

    1. If f(x)=e^{3x^2 + x} find f'(2).

    Solution:

    Differentiate: f(x)=e^{3x^2 + x}

    f'(x)=(6x+1)e^{3x^2 + x}

    Substitute 2 for x.

    Thus, f'(2)=13e^{14}

    2. Find the slope of the tangent to the function: f(x)=2^{x^2+3x when
    x is equal to 3.

    Solution:

    Differentiate: f(x)=2^{x^2+3x}

    f'(x)=2^{x^2+3x} * ln2 * 2x+3

    Thus, f'(3)=(2^{18})(ln2) * 9; f'(3)=1,635,339.371

    Thanks in advance.

    Sincerely,

    Raymond
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: Derivatives of Exponential Functions

    They're both correct!
    Only 1,635,.. what do you mean with this?
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  3. #3
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    Re: Derivatives of Exponential Functions

    Well, I mean that's what the slope of the tangent works out to be if you actually perform the calculations I have indicated in my solution:

    f'(3)=(2^{18})(ln2)(9)
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  4. #4
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    Re: Derivatives of Exponential Functions

    Is it not common practice to fully expand upon one's solution? Because I have been baffled with some large numbers as solutions which seem untypical to the course. Though the numbers, as far as I can tell are accurate. The questions above being one example.
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  5. #5
    MHF Contributor Siron's Avatar
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    Re: Derivatives of Exponential Functions

    Quote Originally Posted by raymac62 View Post
    Well, I mean that's what the slope of the tangent works out to be if you actually perform the calculations I have indicated in my solution:

    f'(3)=(2^{18})(ln2)(9)
    I would just use this answer and don't expand it, but if you want to expand it that's offcourse no problem.
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  6. #6
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    Re: Derivatives of Exponential Functions

    Yeah, I guess that's what I was sort of curious about most with these solutions. I was 99% sure I had nailed the derivatives. I just wasn't sure what the mathematical conventions were in terms of presenting the final solution.

    The first one involving Euler's constant seems very logical that you would leave it as is but I wasn't sure about the other one.
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  7. #7
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    Re: Derivatives of Exponential Functions

    Yes, but I also think you can simplify further with the not expanded form if that would be necessary.
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  8. #8
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    Re: Derivatives of Exponential Functions

    Quote Originally Posted by raymac62 View Post
    Hi,

    I was wondering if someone could check my work for the following two short problems:

    1. If f(x)=e^{3x^2 + x} find f'(2).

    Solution:

    Differentiate: f(x)=e^{3x^2 + x}

    f'(x)=(6x+1)e^{3x^2 + x}

    Substitute 2 for x.

    Thus, f'(2)=13e^{14}

    2. Find the slope of the tangent to the function: f(x)=2^{x^2+3x when
    x is equal to 3.

    Solution:

    Differentiate: f(x)=2^{x^2+3x}

    f'(x)=2^{x^2+3x} * ln2 * 2x+3
    Technical point: Better to use parentheses as you did with the first problem:
    2^{x^2+ 3x}(ln2)(2x+ 3)
    By "precedence of arithmetic operators", what you wrote should be interpreted as
    2^{x^2+ 3x}(ln2)(2x)+ 3

    Thus, f'(3)=(2^{18})(ln2) * 9; f'(3)=1,635,339.371
    In addition to what others have said, not that this is approximately correct because you have rounded of while 2^{x^2+ 3x}(ln 2)(3x+3) is exact. I would not got to additional work to right an answer that is only approximate when I already had the exact answer.

    Thanks in advance.

    Sincerely,

    Raymond
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