1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Having trouble differentiating exponential equations

  1. Apr 24, 2009 #1
    1. The problem statement, all variables and given/known data
    L(t)=15(0.5^(t/26))
    Find the rate of L, when t=60


    2. Relevant equations



    3. The attempt at a solution

    L'(t) = (15/26)(1/2)^(t/26)ln(1/2)
    L'(60)=(15/26)(1/2)^(60/26)ln(1/2)

    = -0.08

    Did I do this right? If I did it wrong, please say where, I am having great trouble understanding this.
     
  2. jcsd
  3. Apr 24, 2009 #2

    Cyosis

    User Avatar
    Homework Helper

    Yes you did it correctly. You say you have great trouble understanding "this". What is "this" exactly, how to differentiate an exponent?
     
  4. Apr 24, 2009 #3
    I guess I was just lucky to be honest, since I did the question with a friend. My problem is applying the chain rule, and knowing where/when to put ln.

    Here's another, that I have not finished yet (don't know how)

    Flow of lava from a volcano is modelled by

    l(t)= 12(2-0.8^t)

    l is dist from crater in km
    t is time in hours

    how fast is the lava travelling down hillside after 4hours,

    well, firstly i need to differentiate l(t), then solve for t

    Can somebody give me hints for what to do with the t? I'm not quite sure..
     
  5. Apr 24, 2009 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    0.8^t=e^(ln(0.8)*t). The derivative of e^(ct) with respect to t is what? Use the chain rule. Powers of constants are just exponentials.
     
  6. Apr 24, 2009 #5

    jhae2.718

    User Avatar
    Gold Member

    Use this derivative shortcut for exponential functions:
    dy/dx a^u=ln(a)*a^u*du //*du is the derivative of u; this is where you use the chain rule
    For a=e, dy/dx e^u=e^u*du

    You are supposed to find the rate at which lava flows down the hill at 4 hours. Since t is the time in hours and l(t) is the distance of the lava from the crater in km, you need
    to find l'(4) to get the rate. It is not necessary to "solve" for t: t=4.


    If you were told the rate and were supposed to find the time, t, you would use algebra to solve for t, much as any other equation. Since l'(t) is an exponential function, you would have to use logarithms and the property that log(a^b)=b log(a), or that log(a)/log(b)=log_b(a).
     
  7. Apr 25, 2009 #6

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    An exponential is about the easiest function to differentiate!

    (ex)'= ex and (ax)'= ax ln(a).

    More generally, by the chain rule, (af(x))'= af(x)ln(a) f'(x).
     
  8. Apr 25, 2009 #7
    l(t)= 12(2-0.8^t)

    so l'(t) = 12(2-0.8^(t) ln 0.8)

    If that's true, it doesn't seem to get me anywhere.
     
  9. Apr 25, 2009 #8

    Hootenanny

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    That's not quite right, why is there still a 2 there?
     
  10. Apr 25, 2009 #9
    Ah yes.

    so l'(t) = 12[-0.8^(t)] ln 0.8)

    ..
    ..

    So when I sub in 4 for t, I get 1.1. Is this correct?
     
  11. Apr 25, 2009 #10

    Hootenanny

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Indeed you do.
     
  12. Apr 25, 2009 #11
    Because the question asked
    "how fast is the lava travelling down hillside after 4hours"

    One more question, just needs clarifying.

    For example if I had (1/2)^(t/138)

    The derivative would be 1/138(1/2)^t/138 ln 1/2 ?

    I'm having trouble differentiating an exponent with a denominator
     
  13. Apr 25, 2009 #12

    Hootenanny

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Apologies, I was looking at your OP.
     
  14. Apr 25, 2009 #13

    jhae2.718

    User Avatar
    Gold Member

    That derivative would be correct. With a denominator, remember the chain rule" dy/dx=dy/dt*dt/dx.

    To put it simply, you multiply the derivative of the function (the a^x*ln(x) ) by the derivative of the exponent.

    So for a derivative with a denominator, you would have a^(x/b)*ln(a)*dx/b.

    Examples: dy/dx 2^(x/20)=2^(x/20)*ln(2)*1/20
    dy/dx 3.4^[(x^2)/85]=3.4^[(x^2)/85]*ln(3.4)*(2x)/85
    dy/dx 3-5^[(x^4)/220]=-5^[(x^4)/220]*ln(5)*[(x^3)/55]

    Does that help?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Having trouble differentiating exponential equations
Loading...