`(d)/(dx)((7x^(2)-22x+4)/(x^(5)))=`

Options

A. `-21x^(-4)8x^(-5)+20x^(-^(-6)`
B. `-(21x^(-4)-8x^(-5)+20x^(-6))`
C. `21x^(-4)-8x^(-5)+20x^(-6)`
D. `21x^(-4)+8x^(-5)+20x^(-6)`

Answer

To differentiate the function \( f(x) = \frac{7x^2 - 22x + 4}{x^5} \) with respect to \( x \), we can use the quotient rule or simplify the expression first. Here, we will simplify the expression first and then differentiate.

### Step-by-Step Solution:

1. **Rewrite the function**:
\[ f(x) = \frac{7x^2}{x^5} - \frac{22x}{x^5} + \frac{4}{x^5} \]
This simplifies to:
\[ f(x) = 7x^{-3} - 22x^{-4} + 4x^{-5} \]

2. **Differentiate each term**:
We will differentiate \( f(x) \) term by term using the power rule \( \frac{d}{dx}(x^n) = nx^{n-1} \).

- For the first term \( 7x^{-3} \):
\[ \frac{d}{dx}(7x^{-3}) = 7 \cdot (-3)x^{-4} = -21x^{-4} \]

- For the second term \( -22x^{-4} \):
\[ \frac{d}{dx}(-22x^{-4}) = -22 \cdot (-4)x^{-5} = 88x^{-5} \]

- For the third term \( 4x^{-5} \):
\[ \frac{d}{dx}(4x^{-5}) = 4 \cdot (-5)x^{-6} = -20x^{-6} \]

3. **Combine the derivatives**:
Now, we combine the results from each differentiation:
\[ f'(x) = -21x^{-4} + 88x^{-5} - 20x^{-6} \]

4. **Final expression**:
We can write the final derivative as:
\[ f'(x) = -21x^{-4} + 88x^{-5} - 20x^{-6} \]

### Final Answer:
\[ f'(x) = -21x^{-4} + 88x^{-5} - 20x^{-6} \]