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Math Help - First and Second Derivatives

  1. #1
    Newbie midsummer's Avatar
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    First and Second Derivatives

    Find the first and second derivatives of the function y=4x/ square root (X+1)

    Thanks!
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by midsummer View Post
    Find the first and second derivatives of the function y=4x/ square root (X+1)

    Thanks!
    f(x)=\frac{4x}{\sqrt{x+1}}=4x \cdot (x+1)^{-1/2} (remember that sqrt(x)=x^(1/2))

    Note that the constant is not important, because [af(x)]'=af'(x) for any constant.

    Apply product rule :
    f'(x)=4 (x+1)^{-1/2}+4x \cdot \tfrac{-1}{2} \cdot (x+1)^{-1/2-1} =4(x+1)^{-1/2}-2x (x+1)^{-3/2}=2(x+1)^{-3/2} \big[2(x+1)-x\big]

    f'(x)=2(x+1)^{-3/2} \big[x+2\big]

    Apply product rule again to get f''(x)
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  3. #3
    MHF Contributor
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    You will have to use the quotient rule for this question.

    The quotient rule is \frac{d}{dx} \frac{f(x)}{g(x)} = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}.

    We have: y = \frac{4x}{\sqrt{x+1}}

    Therefore, y' = \frac{\sqrt{x+1} \cdot 4 - 4x \cdot (1/2)(x+1)^{-1/2}}{x + 1} = \frac{4(x + 1) - 2x}{(x + 1)^{3/2}} = \frac{2x + 4}{(x + 1)^{3/2}}.

    This is the first derivative.

    To obtain the second derivative, you must use the quotient rule to determine the derivative of the first derivative.
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  4. #4
    Newbie midsummer's Avatar
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    thanks...and could you demonstrate how to find the second derivative? I understand the concept and I am trying to simplify (it's been a while since I have done this stuff and need lots of practice). again I really appreciate the help!

    Mariam
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