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Derivative of f(x)=sqrt(7+sqrtx^3)

  1. May 25, 2012 #1
    1. The problem statement, all variables and given/known data

    Determine the derivative of f(x)=√(7+√x^3)

    2. Relevant equations

    Chain rule: f(x)=fog(x) f'(x)=f'(g(x)) x g'(x)

    3. The attempt at a solution

    f(x)=√(7+x^(3/2))

    f'(x)=((½)x^(-½))(7+x^(3/2))((3/2)x^(½))
    =(3.5x^(-½)+½x)((3/2)x^(½))
    =5.25x^0+¾x^½
    =5.25+¾x^½

    That's as far as i got and i feel like it is completely wrong. Any guidance and help is very much appreciated! Thanks in advance.
     
  2. jcsd
  3. May 25, 2012 #2
    Here lies the mistake, chain rule is f(x)=fog(x) f'(x)=f'(g(x)) x g'(x)

    You can take two functions in the given problem, [itex]f(x) = \sqrt{x}[/itex] and [itex]g(x) = 7+\sqrt{x^3}[/itex]

    If you differentiate [itex]\sqrt{7+\sqrt{x^{3}}}[/itex] following the chain rule, you would get the first term f'(g(x)) as,

    [itex]\frac{1}{2\sqrt{g(x)}}[/itex]

    See your mistake? :smile:
     
  4. May 25, 2012 #3
    Ah yes, thank you!
     
  5. May 25, 2012 #4
    So would it now be f'(x)=(1/(2√7+√x^3))((3/2)x^(1/2))?
     
  6. May 25, 2012 #5

    Mark44

    Staff: Mentor

    This is a confusing (to be charitable) use of notation.
    f(x) is not the same as (f o g)(x), and f'(x) is different from f'(g(x)).

    Also, you should not use x to indicate multiplication, especially when you already have a variable named x.

    A better way to write your chain rule formjula would look like this:
    h(x) = f(g(x)) ==> h'(x) = f'(g(x)) ##\cdot## g'(x)
     
  7. May 26, 2012 #6
    Almost....

    It should be, 1/2(√7+√x^3) Parentheses are very important!
     
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