Let's solve the problem step by step.
### Step 1: Find the derivative of [tex]\( f(x) \)[/tex]
We are given the function [tex]\( f(x) = -\frac{7}{-3 + x^4} \)[/tex]. We need to find its derivative [tex]\( f'(x) \)[/tex].
Using the quotient rule:
[tex]\[
f'(x) = \frac{d}{dx} \left( -\frac{7}{-3 + x^4} \right)
\][/tex]
### Step 2: Apply the Quotient Rule
The quotient rule states that for a function [tex]\( g(x) = \frac{u(x)}{v(x)} \)[/tex], the derivative is given by:
[tex]\[
g'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}
\][/tex]
For [tex]\( f(x) = u(x) / v(x) \)[/tex] where [tex]\( u(x) = -7 \)[/tex] and [tex]\( v(x) = -3 + x^4 \)[/tex]:
- [tex]\( u'(x) = 0 \)[/tex] (since [tex]\( u(x) = -7 \)[/tex] is a constant)
- [tex]\( v'(x) = 4x^3 \)[/tex]
Using the quotient rule, we get:
[tex]\[
f'(x) = \frac{0 \cdot (-3 + x^4) - (-7) \cdot 4x^3}{(-3 + x^4)^2} = \frac{28 x^3}{(x^4 - 3)^2}
\][/tex]
Hence, the derivative is:
[tex]\[
f'(x) = \frac{28 x^3}{(x^4 - 3)^2}
\][/tex]
### Step 3: Determine where the denominator is zero
The next step is to find the singularities of [tex]\( f'(x) \)[/tex], i.e., the values of [tex]\( x \)[/tex] where the denominator [tex]\( (x^4 - 3)^2 \)[/tex] is zero.
Solve for [tex]\( x \)[/tex]:
[tex]\[
x^4 - 3 = 0 \implies x^4 = 3 \implies x = \pm \sqrt[4]{3} \text{ or } x = \pm i \sqrt[4]{3}
\][/tex]
So, the roots (or singularities) are:
[tex]\[
x = \sqrt[4]{3},\; x = -\sqrt[4]{3},\; x = i \sqrt[4]{3},\; x = -i \sqrt[4]{3}
\][/tex]
### Step 4: Determine the radius of convergence
The interval of convergence is determined by the distance to the nearest singularity from the center of the power series (in this case centered at [tex]\( 0 \)[/tex]).
The distance from [tex]\( 0 \)[/tex] to each of these roots is [tex]\( \sqrt[4]{3} \)[/tex]. Thus, the radius of convergence [tex]\( R \)[/tex] is [tex]\( \sqrt[4]{3} \)[/tex].
### Step 5: Express the interval of convergence
The interval of convergence is given as [tex]\( (-R, R) \)[/tex]:
[tex]\[
(-\sqrt[4]{3}, \sqrt[4]{3})
\][/tex]
Hence, the exact interval of convergence for [tex]\( f'(x) \)[/tex] is:
[tex]\[
(-\sqrt[4]{3}, \sqrt[4]{3})
\][/tex]
