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Let's work through the given problem using the information provided:

We are given:
- \( f(5) = 1 \)
- \( f'(5) = 6 \)
- \( g(5) = -3 \)
- \( g'(5) = 2 \)

We are asked to find the following derivatives:

### (a) \( (f + g)'(5) \)
By the sum rule of differentiation:
\[
(f + g)'(x) = f'(x) + g'(x)
\]
Thus, at \( x = 5 \):
\[
(f + g)'(5) = f'(5) + g'(5) = 6 + 2 = 8
\]

### (b) \( (fg)'(5) \)
By the product rule of differentiation:
\[
(fg)'(x) = f'(x)g(x) + f(x)g'(x)
\]
Thus, at \( x = 5 \):
\[
(fg)'(5) = f'(5)g(5) + f(5)g'(5) = 6(-3) + 1(2) = -18 + 2 = -16
\]

### (c) \( \left( \frac{f}{g} \right)'(5) \)
By the quotient rule of differentiation:
\[
\left( \frac{f}{g} \right)'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}
\]
Thus, at \( x = 5 \):
\[
\left( \frac{f}{g} \right)'(5) = \frac{f'(5)g(5) - f(5)g'(5)}{g(5)^2} = \frac{6(-3) - 1(2)}{(-3)^2} = \frac{-18 - 2}{9} = \frac{-20}{9}
\]

### (d) \( \left( \frac{g}{f} \right)'(5) \)
Again, by the quotient rule of differentiation:
\[
\left( \frac{g}{f} \right)'(x) = \frac{g'(x)f(x) - g(x)f'(x)}{f(x)^2}
\]
Thus, at \( x = 5 \):
\[
\left( \frac{g}{f} \right)'(5) = \frac{g'(5)f(5) - g(5)f'(5)}{f(5)^2} = \frac{2(1) - (-3)(6)}{1^2} = \frac{2 + 18}{1} = 20
\]

### Summary of Answers:
- (a) \( (f + g)'(5) = 8 \)
- (b) \( (fg)'(5) = -16 \)
- (c) \( \left( \frac{f}{g} \right)'(5) = \frac{-20}{9} \)
- (d) \( \left( \frac{g}{f} \right)'(5) = 20 \)

Would you like more details on any of these steps?

Here are 5 related questions to further explore the concepts:
1. How does the product rule differ from the sum rule in calculus?
2. Can you apply the quotient rule for higher-order derivatives?
3. How would these results change if \( f(5) \) and \( g(5) \) were both zero?
4. What is the general form of the product rule for three functions?
5. How would you differentiate a function raised to a power, such as \( (f(x)^2) \)?

**Tip**: Always check if the denominator in the quotient rule is zero to avoid undefined values.

Suppose f(5) = 1, f'(5) = 6, g(5) = -3, and g'(5) = 2. Find: (a) (f + g)'(5), (b) (fg)'(5), (c) (f/g)'(5), and (d) (g/f)'(5).

Learn how to solve differentiation problems using the sum, product, and quotient rules. This example explores the derivatives of combined functions using key calculus rules, suitable for students in Grades 11-12 or early university calculus.

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