To solve the problem, we need to find the derivative of the function:
[tex]\[ f(x) = \frac{4x^{10} - 10x^6 - 4}{2x^2} \][/tex]
Let's follow through the steps to simplify the function and then find its derivative.
### Step 1: Simplify the Function
First, we simplify the given function:
[tex]\[ f(x) = \frac{4x^{10} - 10x^6 - 4}{2x^2} \][/tex]
We can distribute the denominator [tex]\( 2x^2 \)[/tex] across each term in the numerator:
[tex]\[ f(x) = \frac{4x^{10}}{2x^2} - \frac{10x^6}{2x^2} - \frac{4}{2x^2} \][/tex]
Now simplify each fraction:
[tex]\[ f(x) = \frac{4x^{10}}{2x^2} = 2x^8 \][/tex]
[tex]\[ f(x) = \frac{10x^6}{2x^2} = 5x^4 \][/tex]
[tex]\[ f(x) = -\frac{4}{2x^2} = -\frac{2}{x^2} = -2x^{-2} \][/tex]
Putting it all together, we have:
[tex]\[ f(x) = 2x^8 - 5x^4 - 2x^{-2} \][/tex]
### Step 2: Find the Derivative
To find the derivative [tex]\( f'(x) \)[/tex], we need to differentiate each term:
[tex]\[ f(x) = 2x^8 - 5x^4 - 2x^{-2} \][/tex]
Let's find the derivatives term by term:
1. For [tex]\( 2x^8 \)[/tex]:
[tex]\[ \frac{d}{dx}[2x^8] = 16x^7 \][/tex]
2. For [tex]\( -5x^4 \)[/tex]:
[tex]\[ \frac{d}{dx}[-5x^4] = -20x^3 \][/tex]
3. For [tex]\( -2x^{-2} \)[/tex]:
[tex]\[ \frac{d}{dx}[-2x^{-2}] = 4x^{-3} \][/tex]
Combining all these results, the derivative of [tex]\( f(x) \)[/tex] is:
[tex]\[ f'(x) = 16x^7 - 20x^3 + 4x^{-3} \][/tex]
Therefore, the derivative [tex]\( f'(x) \)[/tex] of the given function is:
[tex]\[
f'(x) = \boxed{(20x^{9} - 30x^{5})/x^2 - 2(2x^{10} - 5x^{6} - 2)/x^3}
\][/tex]
