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Chain Rule W/ Composite Functions

  1. Jan 23, 2016 #1
    1. The problem statement, all variables and given/known data
    If d/dx(f(x)) = g(x) and d/dx(g(x)) = f(x2), then d2/dx2(f(x3)) =
    a) f(x6)
    b) g(x3)
    c) 3x2*g(x3)
    d) 9x4*f(x6) + 6x*g(x3)
    e) f(x6) + g(x3)
    2. Relevant equations


    3. The attempt at a solution
    The answer is D. Since d/dx(f(x)) = g(x), I said that d/dx(f(x3)) should equal 3x2*g(x3), then I took the derivative again and first used product rule so 6x*g(x3) + 3x2*f(x6)*6x5 since you would need to do chain rule again but that doesn't match up with the answer. I've been trying to play around with it for a while and it's just not coming out as the answer.
     
  2. jcsd
  3. Jan 24, 2016 #2

    Samy_A

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    Redo the second derivative. The term 3x2*f(x6)*6x5 looks wrong.
    By the product rule: ##\frac{d}{dx} 3x²g(x³)= 6xg(x³) + 3x² \frac{d}{dx} g(x³)##
     
    Last edited: Jan 24, 2016
  4. Jan 24, 2016 #3
    Here's how I did the second derivative:
    I'm taking the derivative of (3x2)*(g(x3)), and I'm going to let (3x2) = u and (g(x3)) = v
    so the second derivative should look like: d2/dx2(f(x3)) = u'v + uv'
    My first term came out nice: u'v = 6x*g(x3)
    Now for my second term, I said that v' = f(x6)*6x5 since d/dx(g(x)) = f(x2) so I assumed that d/dx(g(x3)) would equal f((x2)3) or similarly, f(x6) and then I used chain rule for f(x6) which is why I added the 6x5 at the end
     
  5. Jan 24, 2016 #4

    Samy_A

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    ##\frac{d}{dx}g(x³) \neq g'(x³)##.
    You are making this too complicated. Try to apply the chain rule when computing ##3x² \frac{d}{dx} g(x³)##.
     
  6. Jan 24, 2016 #5
    Ohhh I see now. I did chain rule on the wrong term. That's why it was coming out funky. Thank you!
     
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