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).