Differentiate the following functions:
13. `(3 x^7 + x^5 - 2x^4 + x-3)/( x^4)`

Answer

To differentiate the function \( f(x) = \frac{3x^7 + x^5 - 2x^4 + x - 3}{x^4} \), we can simplify the function first by dividing each term in the numerator by \( x^4 \).

### Step-by-Step Solution:

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

2. **Rewrite the function**:
\[ f(x) = 3x^3 + x - 2 + \frac{1}{x^3} - \frac{3}{x^4} \]
This can be rewritten as:
\[ f(x) = 3x^3 + x - 2 + x^{-3} - 3x^{-4} \]

3. **Differentiate the function**:
Now we differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx}(3x^3) + \frac{d}{dx}(x) - \frac{d}{dx}(2) + \frac{d}{dx}(x^{-3}) - \frac{d}{dx}(3x^{-4}) \]
Using the power rule \( \frac{d}{dx}(x^n) = nx^{n-1} \):
\[ f'(x) = 3 \cdot 3x^{3-1} + 1 - 0 - 3x^{-4} + 12x^{-5} \]
\[ f'(x) = 9x^2 + 1 - 3x^{-4} + 12x^{-5} \]

4. **Final expression**:
Thus, the derivative of the function is:
\[ f'(x) = 9x^2 + 1 - \frac{3}{x^4} + \frac{12}{x^5} \]