If f'(x) = 2xln(-4(x^2-2.75))+(2x^3)/(x^2-2.75) find the domain.
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
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
To find the domain of the function f'(x), we need to identify any values of x that would make the expression undefined.
In this case, we have two parts in the expression that need to be considered: ln(-4(x^2-2.75)) and (x^2-2.75).
1. Domain of ln(-4(x^2-2.75)):
The natural logarithm function, ln(x), is undefined for negative values and zero. Therefore, for ln(-4(x^2-2.75)) to be defined, -4(x^2-2.75) must be greater than zero.
Solving -4(x^2-2.75) > 0:
x^2-2.75 < 0
x^2 < 2.75
Taking the square root of both sides:
|x| < √2.75
|x| < 1.6583...
Therefore, the domain for ln(-4(x^2-2.75)) is -1.6583 < x < 1.6583.
2. Domain of (x^2-2.75):
The quadratic expression x^2-2.75 is defined for all real numbers x.
Now, let's consider if there are any other restrictions for the domain by looking for values of x that may make the denominator of the second term, (2x^3)/(x^2-2.75), equal to zero.
Setting x^2-2.75 = 0 and solving for x:
x^2 = 2.75
x = ±√2.75
x = ±1.6583...
So, if x = ±1.6583, the denominator becomes zero, making the expression undefined.
Combining all the obtained information, the domain of the function f'(x) is:
-1.6583 < x < 1.6583, excluding x = ±1.6583.