derive this function y=\frac{1}{50}x^{2}-\frac{9}{2}x+475

To derive the given function with respect to \( x \), we will apply the power rule to each term. The function is:

\[
y = \frac{1}{50}x^{2} - \frac{9}{2}x + 475
\]

### Step-by-Step Derivation:

1. **Differentiate the first term:**
   \[
   \frac{d}{dx}\left(\frac{1}{50}x^{2}\right) = \frac{1}{50} \times 2x = \frac{2}{50}x = \frac{1}{25}x
   \]

2. **Differentiate the second term:**
   \[
   \frac{d}{dx}\left(-\frac{9}{2}x\right) = -\frac{9}{2}
   \]

3. **Differentiate the constant term:**
   \[
   \frac{d}{dx}\left(475\right) = 0
   \]

### Derivative of the Function:
Combining these results, the derivative of the function is:

\[
\frac{dy}{dx} = \frac{1}{25}x - \frac{9}{2}
\]

Would you like to delve into any specific details of this derivation, or do you have any other questions?

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Here are 5 related questions for further exploration:

1. How can you find the critical points of this function?
2. What does the derivative tell you about the slope of the original function?
3. How would you determine where this function is increasing or decreasing?
4. How does the derivative change if the coefficient of \( x^2 \) is altered?
5. What is the second derivative of this function, and what does it signify?

**Tip:** When dealing with quadratic functions, the derivative is a linear function, which can help you understand the behavior of the original function (like finding its maximum or minimum).

Learn how to derive the function y = (1/50)x^2 - (9/2)x + 475 with detailed step-by-step differentiation. This problem covers differentiation of quadratic and constant functions using the power rule. Suitable for students in Grades 10-12.

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