Find the second derivative

Mathematics

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To find the second derivative, we will use the result from the previous problem for the first derivative and apply the product rule.

  1. Recall the first derivative.

    Let y=(6x2+1)3(2x−5)4(x3−2x+1)2y = \frac{(6x^2+1)^3 (2x-5)^4}{(x^3-2x+1)^2}.
    The first derivative, dydx\frac{dy}{dx}, was found using logarithmic differentiation:

    dydx=y(36x6x2+1+82x−5−2(3x2−2)x3−2x+1)\frac{dy}{dx} = y \left( \frac{36x}{6x^2+1} + \frac{8}{2x-5} - \frac{2(3x^2-2)}{x^3-2x+1} \right)

  2. Define A(x)A(x) for clarity.

    Let A(x)=36x6x2+1+82x−5−2(3x2−2)x3−2x+1A(x) = \frac{36x}{6x^2+1} + \frac{8}{2x-5} - \frac{2(3x^2-2)}{x^3-2x+1}.
    Then the first derivative can be written as dydx=yA(x)\frac{dy}{dx} = y A(x).

  3. Apply the product rule to find the second derivative.

    The second

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