f(x) = 9*x^2 + 4*x - 20
can you find the derivative of this function with a detailed showing of the work used

To find the derivative of the function \( f(x) = 9x^2 + 4x - 20 \), we'll apply the basic rules of differentiation to each term. Let’s go through the solution step-by-step.

### Step 1: Set Up the Derivative
The function given is:
\[
f(x) = 9x^2 + 4x - 20
\]
To find \( f'(x) \), we need to differentiate each term with respect to \( x \).

### Step 2: Differentiate Each Term
We’ll use the following differentiation rules:
1. **Power Rule**: For any term \( ax^n \), the derivative with respect to \( x \) is \( \frac{d}{dx}(ax^n) = n \cdot ax^{n-1} \).
2. **Constant Rule**: The derivative of a constant is zero.

#### Term-by-Term Differentiation
1. **First term: \( 9x^2 \)**
   \[
   \frac{d}{dx}(9x^2) = 2 \cdot 9 \cdot x^{2-1} = 18x
   \]

2. **Second term: \( 4x \)**
   \[
   \frac{d}{dx}(4x) = 4
   \]

3. **Third term: \( -20 \)**
   \[
   \frac{d}{dx}(-20) = 0
   \]

### Step 3: Combine the Results
Now, we can add up the derivatives of each term:
\[
f'(x) = 18x + 4 + 0
\]

### Final Answer
\[
f'(x) = 18x + 4
\]

So, the derivative of \( f(x) = 9x^2 + 4x - 20 \) is:
\[
f'(x) = 18x + 4
\]

Would you like further details, or do you have any questions?

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Here are five follow-up questions you might explore:
1. What is the second derivative of \( f(x) = 9x^2 + 4x - 20 \)?
2. How would you find the critical points of \( f(x) \) using the derivative \( f'(x) \)?
3. How does the derivative \( f'(x) = 18x + 4 \) relate to the slope of the tangent at any point on \( f(x) \)?
4. What is the value of the derivative \( f'(x) \) at \( x = 0 \) and \( x = 1 \)?
5. How can you use \( f'(x) \) to determine intervals where \( f(x) \) is increasing or decreasing?

**Tip:** When differentiating polynomial functions, apply the power rule to each term individually, and remember that constants disappear since their rate of change is zero.

f(x) = 9*x^2 + 4*x - 20 can you find the derivative of this function with a detailed showing of the work used

Learn how to find the derivative of the function f(x) = 9x^2 + 4x - 20 with a detailed, step-by-step solution. This tutorial covers key calculus concepts, including the power rule and constant rule for differentiation. Suitable for students in Grades 11-12.

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