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Differentiating exponential functions

  1. Nov 13, 2011 #1
    k guys.. i'd like to try to figure out how to get to the derrivatives on my own so really just looking for a right or wrong.



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

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

    f'(2)= e^(3(2)^2+2) * [6(2)+1]
    f'(2)=13e^14 ... correct?


    2) find slope of the tangent to the function f(x) = 2^(x^2+3x) when x=3

    f'(x) = (2x+3) 2^(x^2+3x) ln2

    f'(3)= [2(3)+3] 2^[3^2+(3)(3)] ln 2
    f'(3) = 1 635 339.37 ..... correct?


    3) y = xe^(2x)
    y' = 2x^2e^(2x) + e^(2x) .... correct?


    4) f(x) = 3^x + x^3
    f'(x= 3^x ln 3 + 3x^2 .... correct?


    5) y= e^-(2x+5)
    y' = e^(-2x-5) * (-2) ... correct?


    6) y = e^(3x^2-5x+7)
    y' = e^(3x^2-5x+7) * (6x-5) ... correct?


    If anyone can quickly tell me if these are differentiated correctly it would be very helpful. I'm new to doing this type of work and find myself second guessing most my results.
    thanx for any help.
     
    Last edited: Nov 13, 2011
  2. jcsd
  3. Nov 13, 2011 #2

    SammyS

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    They look fine.

    For #3, you could factor out e2x.

    Added in Edit:

    Well ... Dick is right, there is a problem with #3.
     
    Last edited: Nov 13, 2011
  4. Nov 13, 2011 #3

    Dick

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    I would check (3) again.
     
  5. Nov 20, 2011 #4
    thanx guys... did you mean there's a problem with 3 other than i didnt factor out the e^2x?
     
  6. Nov 20, 2011 #5
    I've been trying to work 3 differently but keep coming up with the same.. could someone show me where I'm going wrong please?
     
  7. Nov 20, 2011 #6

    Dick

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    Your first term in 3 should be x time the derivative of e^(2x). What's the derivative of e^(2x)?
     
  8. Nov 20, 2011 #7
    the derivative of e^(2x) is... e^(2x)*(2x)..correct?
     
  9. Nov 20, 2011 #8
    ohh wait i see.. just 2 not 2x lol
     
  10. Nov 20, 2011 #9
    so the final answer should be y'=e^(2x)*(2x+1) ...?
     
  11. Nov 20, 2011 #10

    Dick

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    You got it.
     
  12. Nov 20, 2011 #11
    thanx a lot. ... I have another question about graphing.. If I'm given a graph of f'(x).. and it asks me to find a possible graph of f(x) and f''(x)... should I know the shape of f''(x) based off of f'(x) or am I just finding a possible graph for that as well?
     
  13. Nov 20, 2011 #12

    Dick

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    To make a sketch of f''(x) just think about what the derivative of f'(x) should look like. f''(x) is just the slope of the curve f'(x). Right? To sketch a graph of f(x) think about what the integral of f'(x) would look like.
     
  14. Nov 20, 2011 #13
    oh ok. So if I'm just given a graph and representing the function and not the equation of the function I'm just showing what the graphs should look like. Not what they actually look like.
    thanx dick... you know, its not true what they say about you... you really dont like up to the name lol
     
  15. Nov 20, 2011 #14

    Dick

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    Well, your sketch of the graph of f''(x) should look pretty close to the graph you would get if you were given an equation. Where f'(x) is increasing, f''(x) better be positive, etc. The sketch of f(x) has a little more variable. Since integration give you a +C arbitrary constant you can shift the graph up or down. You are very welcome. And thanks for the compliment :).
     
  16. Nov 20, 2011 #15
    ok... I'm going to add some pics.. could you let me know if there's anything i should change?
     

    Attached Files:

  17. Nov 20, 2011 #16
    wait the last one is the wrong pic...
     
  18. Nov 20, 2011 #17
    this is h''(x)
     

    Attached Files:

  19. Nov 20, 2011 #18

    Dick

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    If you were going to multiply the last one by (-1), then everything looks ok. And you can shift the first one by any constant up or down, and it would still be ok.
     
  20. Nov 20, 2011 #19

    Dick

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    Yes, ok!
     
  21. Nov 20, 2011 #20
    well the first one is f(x) and it is given. the other two are the ones its asking for...

    thanx again dick
     
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