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Homework Help: Derivative of Composition Functions Problem

  1. Oct 14, 2009 #1
    1. The problem statement, all variables and given/known data

    f and g are differentiable functions that have the following properties:
    i. f(x) < 0 for all values of x
    ii. g(5) = 2

    If h(x)= f(x) / g(x) and h'(x) = f '(x) / g(x), then g(x) = _____?

    2. Relevant equations

    Quotient Rule with f(x) and g(x)

    3. The attempt at a solution
    I have no idea where to begin on this problem.
    Would I first solve the derivative of h(x)?

    so h'(x) = f '(x)*g(x) - f(x)*g'(x) / (g(x))2
    then set that equal to the second given (h'(x))= f'(x)/g(x)?

    Would that be a correct way to start?
  2. jcsd
  3. Oct 14, 2009 #2


    User Avatar
    Homework Helper

    if you mean equate the two expressions
    h'(x) = f '(x)*g(x) - f(x)*g'(x) / (g(x))^2 = f '(x)*g(x)

    then yes, sounds good
  4. Oct 14, 2009 #3
    From there,I have no idea where to go next. would the answer to g(x) be an actual number of a function?
  5. Oct 14, 2009 #4


    Staff: Mentor

    Unless g(x) is a constant function, its formula will never be just a number. g(2), for example, would be a number, but g(x) will be a formula that gives the output for an arbitrary input number x.
  6. Oct 14, 2009 #5


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    Homework Helper

    so if h'(x) = f '(x)*g(x) - f(x)*g'(x) / (g(x))^2 = f '(x)*g(x)

    h'(x) - f '(x)*g(x) = - f(x)*g'(x) / (g(x))^2 = 0

    so the term f(x)*g'(x) / (g(x))^2 is zero for all x, you know f(x) <0 for all x, so this is always non-zero, g(5)=2 so this is non-zero for at least one x, what does this tell you about g'(x)
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