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Chain rule, I think (I'm so confused)

  1. Oct 23, 2005 #1
    I'm so confused. I have to find the derivative of f(x) = x^5(4^(x^2)). All of the powers are messing me up. Any help would be much appreciated. Thanks!
     
  2. jcsd
  3. Oct 23, 2005 #2
    Is your function [tex]f(x) = x^5(4^{x^2})[/tex] or [tex]f(x) = x^{5(4^{x^2})}[/tex] ?
     
    Last edited: Oct 23, 2005
  4. Oct 23, 2005 #3
    The first one, sorry.
     
  5. Oct 23, 2005 #4
    Use product rule first, then you end up differentiating 4^(x^2).
    A nice formula to know is d/dx ( a^(f(x)) ) = a^f(x) * ln(a) * f'(x), which comes from the chain rule.
     
  6. Oct 23, 2005 #5
    I know I'm having some sort of stupid lapse right now, but the part I can't figure out is the 4^(x^2).
     
  7. Oct 23, 2005 #6

    HallsofIvy

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    To differentiate [tex]y= 4^{x^2}[/tex], take the logarithm of both sides:
    [tex]ln y= x^2 ln 4[/tex]
    Now differentiate that, with respect to x.
    [tex]\frac{1}{y}y'= 2x ln 4[/tex]
    so
    [tex]y'= 2x (ln 4)y= 2x 4^{x^2} ln 4[/tex]

    Karlsen used the fact that the derivative of ax is ax ln a, but not everyone knows that!
     
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