Let's solve each part step by step using the given table values and derivative rules.

Given Table:

At $x = 2$:

• $f(2) = 8$
• $g(2) = 2$
• $f'(2) = 2\pi$
• $g'(2) = e$

Part a: $\frac{d}{dx} [f(g(x))]$ at $x = 2$

To solve this, we use the chain rule: $\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)$

1. At $x = 2$, $g(2) = 2$, so $f'(g(2)) = f'(2) = 2\pi$.
2. $g'(2) = e$.

Now multiply: $\frac{d}{dx} [f(g(x))] = f'(g(2)) \cdot g'(2) = (2\pi) \cdot e = 2\pi e$

Answer for Part a: $2\pi e$

Part b: $\frac{d}{dx} \left[ \frac{1}{f(x)} \right]$ at $x = 2$

To solve this, use the derivative of a reciprocal rule: $\frac{d}{dx} \left[ \frac{1}{f(x)} \right] = -\frac{f'(x)}{(f(x))^2}$

1. At $x = 2$, $f(2) = 8$ and $f'(2) = 2\pi$.

Substitute these values: $\frac{d}{dx} \left[ \frac{1}{f(x)} \right] = -\frac{f'(2)}{(f(2))^2} = -\frac{2\pi}{(8)^2}$

Simplify: $\frac{d}{dx} \left[ \frac{1}{f(x)} \right] = -\frac{2\pi}{64} = -\frac{\pi}{32}$

Answer for Part b: $-\frac{\pi}{32}$

Part c: $\frac{d}{dx} \left[ \frac{g(x)}{f(x)} \right]$ at $x = 2$

To solve this, use the quotient rule: $\frac{d}{dx} \left[ \frac{g(x)}{f(x)} \right] = \frac{g'(x)f(x) - g(x)f'(x)}{(f(x))^2}$

1. At $x = 2$:
   • $g(2) = 2$, $f(2) = 8$,
   • $g'(2) = e$, $f'(2) = 2\pi$.

Substitute these values: $\frac{d}{dx} \left[ \frac{g(x)}{f(x)} \right] = \frac{g'(2)f(2) - g(2)f'(2)}{(f(2))^2}$ $\frac{d}{dx} \left[ \frac{g(x)}{f(x)} \right] = \frac{(e)(8) - (2)(2\pi)}{(8)^2}$

Simplify the numerator: $(e)(8) - (2)(2\pi) = 8e - 4\pi$

Simplify the denominator: $(8)^2 = 64$

Thus: $\frac{d}{dx} \left[ \frac{g(x)}{f(x)} \right] = \frac{8e - 4\pi}{64}$

Simplify further: $\frac{d}{dx} \left[ \frac{g(x)}{f(x)} \right] = \frac{8e}{64} - \frac{4\pi}{64} = \frac{e}{8} - \frac{\pi}{16}$

Answer for Part c: $\frac{e}{8} - \frac{\pi}{16}$

Final Answers:

a) $2\pi e$
b) $-\frac{\pi}{32}$
c) $\frac{e}{8} - \frac{\pi}{16}$

Do you need any further clarifications or explanations? Let me know!

Here are 5 related questions for practice:

1. Differentiate $h(x) = f(x) \cdot g(x)$ at $x = 2$.
2. Find the derivative of $f(g(h(x)))$ using the chain rule.
3. Evaluate $\frac{d}{dx} \left[ x \cdot f(x) \right]$ at $x = 2$.
4. Differentiate $\frac{1}{g(x)}$ at $x = 2$.
5. Find $\frac{d}{dx} \left[ g(x)^2 \right]$ at $x = 2$.

Tip: Always identify the derivative rule (chain, product, or quotient) before solving.