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The image you uploaded shows exercises for finding the derivative of functions using the limit process. I will help you find the derivatives of the functions listed in exercises 11 through 20.

### Derivatives of the functions:

1. **Exercise 11: \( f(x) = 4 \)**  
   Since this is a constant function, the derivative is:
   \[
   f'(x) = 0
   \]

2. **Exercise 12: \( g(x) = -8 \)**  
   Another constant function, so its derivative is:
   \[
   g'(x) = 0
   \]

3. **Exercise 13: \( f(x) = -5x \)**  
   The derivative of a linear function \( ax \) is simply the coefficient \( a \). Thus:
   \[
   f'(x) = -5
   \]

4. **Exercise 14: \( f(x) = 6x - 2 \)**  
   The derivative of \( 6x \) is 6, and the derivative of the constant \( -2 \) is 0:
   \[
   f'(x) = 6
   \]

5. **Exercise 15: \( f(x) = x^2 - 3 \)**  
   The derivative of \( x^2 \) is \( 2x \), and the derivative of the constant \( -3 \) is 0:
   \[
   f'(x) = 2x
   \]

6. **Exercise 16: \( f(x) = x^2 + x - 5 \)**  
   The derivative of \( x^2 \) is \( 2x \), the derivative of \( x \) is 1, and the derivative of the constant \( -5 \) is 0:
   \[
   f'(x) = 2x + 1
   \]

7. **Exercise 17: \( f(x) = x^3 - 12x \)**  
   The derivative of \( x^3 \) is \( 3x^2 \), and the derivative of \( -12x \) is \( -12 \):
   \[
   f'(x) = 3x^2 - 12
   \]

8. **Exercise 18: \( g(t) = t^3 + 4t \)**  
   The derivative of \( t^3 \) is \( 3t^2 \), and the derivative of \( 4t \) is 4:
   \[
   g'(t) = 3t^2 + 4
   \]

9. **Exercise 19: \( f(x) = \frac{1}{x - 1} \)**  
   This is a rational function, so we apply the chain rule:
   \[
   f'(x) = -\frac{1}{(x - 1)^2}
   \]

10. **Exercise 20: \( f(x) = \frac{1}{x^2} \)**  
    This can be rewritten as \( f(x) = x^{-2} \), and we apply the power rule:
    \[
    f'(x) = -2x^{-3} = -\frac{2}{x^3}
    \]

These are the derivatives for each function. Do you want further clarification on any of these steps?

### Related Questions:
1. What is the formal definition of the derivative using the limit process?
2. How does the power rule apply to both positive and negative exponents?
3. Can the chain rule be applied to more complex rational functions?
4. What is the product rule and how does it differ from the quotient rule?
5. How does the derivative of \( e^x \) compare to polynomial derivatives?

#### Tip:
Always simplify functions, when possible, before applying differentiation rules, as it often makes the process easier!

In Exercises 11-20, find the derivative of the function by the limit process. Functions include constants, linear functions, quadratic, cubic, and rational functions.

Find the derivatives of various functions using the limit process. Includes constant, linear, quadratic, cubic, and rational functions. Suitable for high school calculus students.

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