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Math Help - Higher Derivatives

  1. #1
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    Higher Derivatives

    1) Find the first and second derivatives:

    y = Square Root((x^2) + 1)

    2) If f(x) = Square Root(1 + (x^3)) , find the second derivative (2). So x = 2.
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  2. #2
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    y= (x^2+ 1)^{1/2}

    Apply the power rule, (u^n)'= nu^{n-1} and the chain rule.
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  3. #3
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    Did that and got the wrong answer
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  4. #4
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    Then show what you did, please.
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  5. #5
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    First Deriv. = (1/2)(x^2 + 1) ^ (-1/2) (2x)
    = x(x^2 + 1) ^ (-1/2)

    Second = (-1/2x) (x^2 + 1) ^ (-3/2) (2x)

    = -x^2 (x^2 + 1) ^ (-3/2)

    Book says it should be only an x in the front
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  6. #6
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    Quote Originally Posted by Dickson View Post
    First Deriv. = (1/2)(x^2 + 1) ^ (-1/2) (2x)
    = x(x^2 + 1) ^ (-1/2)

    Second = (-1/2x) (x^2 + 1) ^ (-3/2) (2x)

    = -x^2 (x^2 + 1) ^ (-3/2)

    Book says it should be only an x in the front
    To get the second derivative you have to use either the product or quotient rule as well as the chain rule.

    I suggest writing the first derivative as \frac{x}{(x^2 + 1)^{1/2}} and using the quotient rule to differentiate. To get the derivative of the denominator the chain rule is required.

    A bit of algebra will be required to get the answer in the required form.
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