https://file.aiquickdraw.com/mathbot/file/1727661435382uk1edzm9.jpg

The given function is \( f(x) = (1 + 2x)(5 - 2x) \), and you are asked to:

1. Find the derivative \( f'(x) \).
2. Find the equation of the tangent line to the graph of \( f \) at \( x = 2 \).

### Step 1: Finding \( f'(x) \)
To differentiate \( f(x) = (1 + 2x)(5 - 2x) \), we apply the **product rule**.

The product rule states:
\[
(fg)' = f'g + fg'
\]
Where \( f(x) = (1 + 2x) \) and \( g(x) = (5 - 2x) \).

First, we compute the derivatives of each factor:

- \( \frac{d}{dx}(1 + 2x) = 2 \)
- \( \frac{d}{dx}(5 - 2x) = -2 \)

Now, apply the product rule:
\[
f'(x) = (1 + 2x)(-2) + (5 - 2x)(2)
\]
Simplifying:
\[
f'(x) = -2(1 + 2x) + 2(5 - 2x)
\]
\[
f'(x) = -2 - 4x + 10 - 4x
\]
\[
f'(x) = 8 - 8x
\]

This matches the derivative shown in the image: \( f'(x) = 8 - 8x \).

### Step 2: Finding the equation of the tangent line
The equation of the tangent line at \( x = 2 \) can be written in point-slope form:
\[
y - f(2) = f'(2)(x - 2)
\]
First, calculate \( f'(2) \):
\[
f'(2) = 8 - 8(2) = 8 - 16 = -8
\]

Now, calculate \( f(2) \):
\[
f(2) = (1 + 2(2))(5 - 2(2)) = (1 + 4)(5 - 4) = 5(1) = 5
\]

So, the equation of the tangent line is:
\[
y - 5 = -8(x - 2)
\]
Simplify:
\[
y - 5 = -8x + 16
\]
\[
y = -8x + 21
\]

Thus, the equation of the tangent line is:
\[
y = -8x + 21
\]

Would you like further details or have any questions about these steps?

### Relative questions:
1. How does the product rule apply to more complex functions?
2. What is the geometric meaning of a tangent line in calculus?
3. Can you apply the quotient rule to this function instead of the product rule?
4. How do you find the normal line to a curve at a specific point?
5. How would this problem change if \( f(x) \) was a trigonometric function?

### Tip:
Always check both the slope and the point on the curve when deriving the equation of a tangent line to ensure accuracy.

Find f'(x) and find the equation of the line tangent to the graph of f at x=2. f(x) = (1 + 2x)(5 - 2x)

Learn how to differentiate the function f(x) = (1 + 2x)(5 - 2x) using the product rule and find the equation of the tangent line at x = 2. This solution covers step-by-step differentiation and application of the point-slope form for the tangent line. Suitable for high school students in Grades 11-12.

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