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.