find the derivative of f(x)=(x-9.5)^3+(x-9)+20{9<x<15}

To find the derivative of the function \( f(x) = (x - 9.5)^3 + (x - 9) + 20 \), we will differentiate each term with respect to \( x \).

Given:
\[
f(x) = (x - 9.5)^3 + (x - 9) + 20
\]

### Step 1: Differentiate the first term \( (x - 9.5)^3 \)
The derivative of \( (x - 9.5)^3 \) with respect to \( x \) is obtained using the chain rule:
\[
\frac{d}{dx}\left[(x - 9.5)^3\right] = 3(x - 9.5)^2 \cdot \frac{d}{dx}(x - 9.5) = 3(x - 9.5)^2
\]

### Step 2: Differentiate the second term \( (x - 9) \)
The derivative of \( (x - 9) \) with respect to \( x \) is:
\[
\frac{d}{dx}\left[(x - 9)\right] = 1
\]

### Step 3: Differentiate the third term \( 20 \)
The derivative of a constant is 0:
\[
\frac{d}{dx}\left[20\right] = 0
\]

### Step 4: Combine the results
The derivative of \( f(x) \) is:
\[
f'(x) = 3(x - 9.5)^2 + 1
\]

So, the derivative of \( f(x) \) is:
\[
f'(x) = 3(x - 9.5)^2 + 1
\]

### Domain Consideration
The function is defined on the interval \( 9 < x < 15 \), but this does not affect the differentiation process.

Would you like more details or have any further questions?

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Here are 5 related questions to explore:

1. What is the second derivative of the function \( f(x) \)?
2. How would you find the critical points of \( f(x) \)?
3. What does the derivative \( f'(x) \) tell us about the behavior of \( f(x) \) on the interval \( 9 < x < 15 \)?
4. Can you find the tangent line to \( f(x) \) at a specific point, say \( x = 10 \)?
5. How does the derivative of a polynomial function change with respect to its degree?

**Tip:** Always remember to apply the chain rule correctly when differentiating composite functions.

Learn how to find the derivative of the function f(x) = (x - 9.5)^3 + (x - 9) + 20 step-by-step. This problem covers differentiation techniques, including the chain rule and derivatives of polynomials. Suitable for advanced high school students.

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