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This image presents an example from a textbook on finding the derivative of a function and using it to find the equation of a tangent line.

### Problem:
1. Find the derivative of \( f(x) = x^2 \) with respect to \( x \).
2. Use the derivative to find the equation of the tangent line to \( y = x^2 \) at \( x = 2 \).

### Solution:
1. **Derivative Calculation**:
   The derivative of a function using the definition is given by:
   \[
   f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
   \]
   Substituting \( f(x) = x^2 \):
   \[
   f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h}
   \]
   Expanding \( (x+h)^2 \):
   \[
   f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h}
   \]
   Simplifying:
   \[
   f'(x) = \lim_{h \to 0} \left( 2x + h \right) = 2x
   \]
   Thus, the derivative of \( f(x) = x^2 \) is:
   \[
   f'(x) = 2x
   \]

2. **Tangent Line at \( x = 2 \)**:
   To find the equation of the tangent line, we use the point-slope form of the equation of a line:
   \[
   y - y_1 = m(x - x_1)
   \]
   Here, \( m = f'(2) = 2(2) = 4 \), and the point of tangency is \( (2, f(2)) = (2, 4) \).

   Substituting these values:
   \[
   y - 4 = 4(x - 2)
   \]
   Simplifying:
   \[
   y = 4x - 8 + 4 \quad \Rightarrow \quad y = 4x - 4
   \]

Thus, the equation of the tangent line is:
\[
y = 4x - 4
\]

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Would you like more details or have any questions?

Here are five related questions:
1. How do you differentiate higher powers of \( x \), like \( f(x) = x^3 \)?
2. What is the geometric interpretation of a derivative?
3. How does the tangent line change if the point of tangency changes?
4. Can you find the second derivative of \( f(x) = x^2 \) and explain its meaning?
5. How would the derivative change if the function was \( f(x) = x^2 + 3x \)?

**Tip:** The derivative of a function gives the slope of the tangent line to the curve at any point.

Find the derivative of f(x) = x^2 with respect to x and use it to find the equation of the tangent line to y = x^2 at x = 2.

This problem covers how to find the derivative of f(x) = x^2 using the limit definition and use it to find the equation of the tangent line at x = 2. Key concepts include derivatives, limits, and the tangent line equation. Suitable for students in Grades 10-12.

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