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Let's solve the derivatives of each transformation given in the table step by step:

### (a) \( g(x) = f(x) - 2 \)

- Since \( g(x) = f(x) - 2 \), the derivative is \( g'(x) = f'(x) \). This is because the derivative of a constant (\(-2\)) is zero.
- Thus, the values for \( g'(x) \) will be the same as those for \( f'(x) \):
  - \( g'(-2) = f'(-2) = 6 \)
  - \( g'(-1) = f'(-1) = \frac{1}{3} \)
  - \( g'(0) = f'(0) = -\frac{2}{3} \)
  - \( g'(1) = f'(1) = -1 \)
  - \( g'(2) = f'(2) = -4 \)
  - \( g'(3) = f'(3) = -6 \)

### (b) \( h(x) = 2f(x) \)

- For \( h(x) = 2f(x) \), the derivative is \( h'(x) = 2f'(x) \).
- Multiply each \( f'(x) \) value by 2:
  - \( h'(-2) = 2 \cdot 6 = 12 \)
  - \( h'(-1) = 2 \cdot \frac{1}{3} = \frac{2}{3} \)
  - \( h'(0) = 2 \cdot -\frac{2}{3} = -\frac{4}{3} \)
  - \( h'(1) = 2 \cdot -1 = -2 \)
  - \( h'(2) = 2 \cdot -4 = -8 \)
  - \( h'(3) = 2 \cdot -6 = -12 \)

### (c) \( r(x) = f(-3x) \)

- For \( r(x) = f(-3x) \), use the chain rule: \( r'(x) = f'(-3x) \cdot (-3) \).
- To find \( r'(x) \) at specific values, substitute \( -3x \) into \( f'(x) \):
  - \( r'(-2) = f'(6) \cdot (-3) = \text{undefined} \) (because \( f'(6) \) is not given)
  - \( r'(-1) = f'(3) \cdot (-3) = -6 \cdot (-3) = 18 \)
  - \( r'(0) = f'(0) \cdot (-3) = -\frac{2}{3} \cdot (-3) = 2 \)
  - \( r'(1) = f'(-3) \cdot (-3) = \text{undefined} \) (because \( f'(-3) \) is not given)
  - \( r'(2) = f'(-6) \cdot (-3) = \text{undefined} \) (because \( f'(-6) \) is not given)
  - \( r'(3) = f'(-9) \cdot (-3) = \text{undefined} \) (because \( f'(-9) \) is not given)

### (d) \( s(x) = f(x + 2) \)

- For \( s(x) = f(x + 2) \), use the chain rule: \( s'(x) = f'(x + 2) \).
- Shift the input values by 2:
  - \( s'(-2) = f'(0) = -\frac{2}{3} \)
  - \( s'(-1) = f'(1) = -1 \)
  - \( s'(0) = f'(2) = -4 \)
  - \( s'(1) = f'(3) = -6 \)
  - \( s'(2) = f'(4) = \text{undefined} \) (because \( f'(4) \) is not given)
  - \( s'(3) = f'(5) = \text{undefined} \) (because \( f'(5) \) is not given)

Would you like a more detailed explanation of any part of these calculations?

### Related Questions:
1. What is the chain rule, and how is it applied in derivative transformations?
2. How does multiplying a function by a constant affect its derivative?
3. Why does shifting the input of a function affect its derivative in a particular way?
4. What happens to the derivative when composing functions like \( f(g(x)) \)?
5. How do undefined points in the derivative table affect function analysis?

### Tip:
When using the chain rule, always remember to multiply the derivative of the inner function by the derivative of the outer function for accurate results!

The table shows some values of the derivative of an unknown function f. Complete the table by finding the derivative of each transformation of f, if possible. (If an answer is undefined, enter UNDEFINED.)

Solve the derivatives of various transformations of an unknown function f using rules like the chain rule and constant multiple rule. This problem involves computing derivatives of shifted functions, composed functions, and constant multiples of f. Learn step-by-step solutions for each case.

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