To interpret your question correctly, it seems you want the simplified forms or a summary of each derivative expression in fractional form.

Let's review and simplify each expression provided:

1. **Expression:**  
   \[
   \frac{dy}{dx} = -\frac{1}{30}x - \frac{13}{3}
   \]
   - This expression is already in its simplest fractional form.

2. **Expression:**  
   \[
   \frac{dy}{dx} = -\frac{x^2}{840} + \frac{38x}{63} - \frac{4370}{63}
   \]

   - To simplify, each term needs to be expressed with a common denominator.
   - Let's simplify this expression step-by-step:
   
   \[
   \frac{dy}{dx} = -\frac{x^2}{840} + \frac{38x}{63} - \frac{4370}{63}
   \]

   - The common denominator for \(840\) and \(63\) is \(2520\):
     \[
     -\frac{x^2}{840} = -\frac{3x^2}{2520}, \quad \frac{38x}{63} = \frac{1520x}{2520}, \quad -\frac{4370}{63} = -\frac{17480}{2520}
     \]
   
   - Combining these fractions:
     \[
     \frac{dy}{dx} = -\frac{3x^2}{2520} + \frac{1520x}{2520} - \frac{17480}{2520} = \frac{-3x^2 + 1520x - 17480}{2520}
     \]

3. **Expression:**  
   \[
   \frac{dy}{dx} = -5.033218 \cdot \sin(0.419x - 130.649)
   \]

   - This expression involves a sinusoidal function. The numerical coefficient can be left as is unless you're looking for a rational approximation or fractional representation.
   
Given these derivatives, the final simplified fractional form for the quadratic expression is:

\[
\frac{dy}{dx} = \frac{-3x^2 + 1520x - 17480}{2520}
\]

### Summary of Simplified Fractions

1. **Linear Function:**  
   \[
   \frac{dy}{dx} = -\frac{1}{30}x - \frac{13}{3}
   \]

2. **Quadratic Function:**  
   \[
   \frac{dy}{dx} = \frac{-3x^2 + 1520x - 17480}{2520}
   \]

3. **Sinusoidal Function:**  
   This remains as it is:
   \[
   \frac{dy}{dx} = -5.033218 \cdot \sin(0.419x - 130.649)
   \]

Would you like to see specific evaluations or have more details on any of these?

### Questions for Further Exploration

1. How do you interpret the simplified fractions for practical use, such as finding slopes at specific points?
2. What does each coefficient and term represent in the context of these equations?
3. How can these functions be visualized graphically to better understand their behavior?
4. Are there specific points (like where \(x = 0\)) that make evaluating these derivatives easier?
5. How do different types of functions (linear, quadratic, sinusoidal) affect their derivatives?

**Tip:** When simplifying fractions in algebraic expressions, finding a common denominator is crucial to combining terms effectively.

Explore how to simplify derivatives into fractional forms with detailed examples. Learn about derivatives of linear, quadratic, and sinusoidal functions and their fractional representations. Suitable for advanced high school students.

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