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Let's address the questions in the image.

The function given is \( f(x) = x + |x| \).

### Part (a): Using the definition of the derivative, find \( f'(2) \).

The definition of the derivative is:

\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]

For \( f(x) = x + |x| \), we must first understand how the absolute value affects the derivative. The function behaves differently depending on whether \( x \) is positive or negative:

- For \( x \geq 0 \), \( f(x) = x + x = 2x \).
- For \( x < 0 \), \( f(x) = x - x = 0 \).

Since \( x = 2 \) is positive, the derivative will be calculated using the expression for \( x \geq 0 \), i.e., \( f(x) = 2x \).

Now, applying the definition of the derivative at \( x = 2 \):

\[
f'(2) = \lim_{h \to 0} \frac{f(2+h) - f(2)}{h}
\]

Substituting \( f(2+h) = 2(2+h) = 4 + 2h \) and \( f(2) = 4 \):

\[
f'(2) = \lim_{h \to 0} \frac{(4 + 2h) - 4}{h} = \lim_{h \to 0} \frac{2h}{h} = 2
\]

Thus, \( f'(2) = 2 \).

### Part (b): Find an equation of the tangent line to \( f \) at \( x = 2 \).

The equation of a tangent line is given by:

\[
y - f(a) = f'(a)(x - a)
\]

We already know that \( f'(2) = 2 \) and we can find \( f(2) \) using the function \( f(x) = 2x \) for \( x \geq 0 \):

\[
f(2) = 2(2) = 4
\]

Therefore, the equation of the tangent line at \( x = 2 \) is:

\[
y - 4 = 2(x - 2)
\]

Simplifying:

\[
y = 2x - 4 + 4 \implies y = 2x
\]

So, the equation of the tangent line is \( y = 2x \).

---

Would you like further clarification on any of these steps? Here are 5 related questions for further understanding:

1. How does the definition of the derivative work when applied to non-linear functions?
2. What would happen if we calculated the derivative at \( x = -2 \)?
3. Can you describe the behavior of \( f(x) = x + |x| \) for negative values of \( x \)?
4. How does the absolute value affect the differentiability of a function?
5. How would the tangent line change if \( x = 3 \) instead of \( x = 2 \)?

**Tip:** When working with absolute value functions, always consider the behavior of the function separately in different regions (positive and negative parts).

Let f(x) = x + |x|. (a) Using the definition of the derivative, find f'(2). (b) Find an equation of the tangent line to f at x = 2.

Learn how to find the derivative of f(x) = x + |x| at x = 2 using the definition of the derivative and the equation of the tangent line. This problem involves key calculus concepts such as derivatives, absolute value functions, and tangent lines. Suitable for students in Grades 11-12.

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