CALCULUS HELP?

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  • 2 months ago
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    f(x) = x² / (x + 2)

    to get this first derivative we need to use the quotient rule:

    if f = u/v, then f' = (u'v - uv') / v²

    So we can set up our function to be:

    f(x) = u/v and u = x² and v = x + 2

    Now we can get the first derivative of u and v:

    u' = 2x and v' = 1

    we can now substsitute what we know to get f'(x):

    f'(x) = (u'v - uv') / v²

    f'(x) = [2x(x + 2) - x²(1)] / (x + 2)²

    Simplify to get:

    f'(x) = (2x² + 4x - x²) / (x + 2)²

    f'(x) = (x² + 4x) / (x + 2)²

    Now we substitute and solve for the two values you are looking for:

    f'(0) = (0² + 4 * 0) / (0 + 2)² and f'(21) = (21² + 4 * 21) / (21 + 2)²

    f'(0) = (0 + 0) / 2² and f'(21) = (441 + 84) / 23²

    f'(0) = 0 / 4 and f'(21) = 525 / 529

    f'(0) = 0 and f'(21) = 525/529

    Those are your answers.

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