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If f'(x) = 2xln(-4(x^2-2.75))+(2x^3)/(x^2-2.75) find the domain.

  • Calculus -

    remember we can only take logs (or ln) of positive numbers, so

    -4(x^2 - 2.75) > 0 also for 2x^3/(x^2 - 2.75), the denominator cannot be zero
    But we already included that in our first restriction

    -4(x^2 - 2.75) > 0
    x^2 - 2.75 < 0

    x^2 < 2.75 --- > x^2 < 11/4

    - √11/2 < x < √11/2

  • Calculus -

    since ln(x) has domain x>0, we need

    -4(x^2-2.75) > 0
    -√11/2 < x < √11/2

    Note that this also works for the 2nd term, since it must be nonzero. That is,

    x ≠ √11/2

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