The image contains a list of exercises that ask to find the derivative of various functions using the limit process. Let's walk through the steps for a few examples:

Example 11:

$f(x) = 7$

Since this is a constant function, its derivative is zero: $f'(x) = 0$

Example 13:

$f(x) = -10x$

For a linear function $f(x) = ax$, the derivative is simply the constant coefficient $a$. Thus: $f'(x) = -10$

Example 17:

$f(x) = x^2 + x - 3$

Using the sum and power rules of differentiation: $f'(x) = 2x + 1$

Example 19:

$f(x) = x^3 - 12x$

Using the power rule: $f'(x) = 3x^2 - 12$

Example 21:

$f(x) = \frac{1}{x - 1}$

This is a rational function. We rewrite it as $f(x) = (x - 1)^{-1}$, and then apply the power rule: $f'(x) = -(x - 1)^{-2}$ Or equivalently: $f'(x) = -\frac{1}{(x - 1)^2}$

Would you like details on any specific problem or steps involved? Here are some further questions that can deepen your understanding:

1. How is the limit definition of a derivative applied to these problems?
2. How does the power rule simplify derivative calculations?
3. Why does the derivative of a constant equal zero?
4. What is the chain rule, and how might it apply to more complex functions?
5. How would you compute the derivative of a product or quotient of two functions?

Tip: Understanding the power rule will help you with most polynomial functions and save time in differentiation.