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

Answer 1

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

Answer 2

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