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: $y = 4x - 4$

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.