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Chain rule question: if f is a differentiable function

  1. Feb 27, 2013 #1
    If f is a differntiable function, find the expression for derivatives of the following functions.

    a) g(x)= x/ f(x)
    b) h(x) [f(x^3)]^2
    c) k(x)= sqrt (1 + [f(x)]^2)

    First off, I am not even sure what they are asking. I am assuming that they want the derivative for each component of the equation? then to find the derivative for the entire function?

    a) really not sure about this one

    b) g(x) = x^2 f(x)= x^3
    g'(x)= 2x f'(x)= 3x^2
    h'(x)= 2(x^3)(3x^2)
    h'(x)= (2x^3)(3x^2)
    h'(x)= 6x^5

    c) g(x)= sqrt (x) h(x)= 1 + x^2
    g'(x)= 1/2 x^-1/2 h'(x)= 2x
    f'(x)= 1
    k'(x)= 1/2 1 + (x^2)^-1/2(2x)
    then continue to find equation.


    The fact that f(x) is in the equation is throwing me off. Can you explain why you are approaching the problem this way. I am doing my best but we were given this yesterday to solve, but without understanding the question, I am a little at a loss. Thank you so much!!
     
  2. jcsd
  3. Feb 27, 2013 #2
    Well for the first one just think of it as any function. f(x) can be x^2, x^3, etc.

    So use the quotient rule:
    [tex]f'(x)=\frac{g'(x)h(x)-g(x)h'(x)}{(h(x))^{2}}[/tex]
     
  4. Feb 27, 2013 #3
    For the second one I would use the chain rule ie bring the square down and the 3x^2 out of the inside of the function to obtain (6x^2)(f(x^3)) I think :-)
     
  5. Feb 27, 2013 #4

    CAF123

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    Gold Member

    For the second one, to take into account f is a function of x^3, you should use the chain rule again to differentiate with respect to x^3.

    For the third one, you have to apply the chain rule multiple times.
     
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