It looks like you're working on a calculus worksheet involving derivatives by first principles. I'll guide you through solving these problems.

First principles formula:

The derivative $f'(x)$ by first principles is given by the limit:

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Now let's approach both parts:

Part (a)

The function provided is:

$f(x) = \frac{1}{x - 3}$

Using the first principles, we need to evaluate:

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Substitute $f(x)$ and $f(x+h)$:

$f'(x) = \lim_{h \to 0} \frac{\frac{1}{(x+h)-3} - \frac{1}{x-3}}{h}$

Simplify the numerator:

$= \lim_{h \to 0} \frac{\frac{1}{x+h-3} - \frac{1}{x-3}}{h} = \lim_{h \to 0} \frac{(x-3)-(x+h-3)}{(x+h-3)(x-3)} \cdot \frac{1}{h}$

$= \lim_{h \to 0} \frac{-h}{(x+h-3)(x-3) \cdot h}$

Cancel the $h$ in the numerator and denominator:

$= \lim_{h \to 0} \frac{-1}{(x+h-3)(x-3)}$

As $h \to 0$, the expression simplifies to:

$f'(x) = \frac{-1}{(x-3)^2}$

Part (b)

The function provided is:

$f(x) = \frac{1}{x + 2}$

Again, using the first principles:

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Substitute $f(x)$ and $f(x+h)$:

$f'(x) = \lim_{h \to 0} \frac{\frac{1}{x+h+2} - \frac{1}{x+2}}{h}$

Simplify the numerator:

$= \lim_{h \to 0} \frac{(x+2)-(x+h+2)}{(x+h+2)(x+2)} \cdot \frac{1}{h}$

$= \lim_{h \to 0} \frac{-h}{(x+h+2)(x+2) \cdot h}$

Cancel the $h$:

$= \lim_{h \to 0} \frac{-1}{(x+h+2)(x+2)}$

As $h \to 0$, this simplifies to:

$f'(x) = \frac{-1}{(x+2)^2}$

Final answers:

• (a) $f'(x) = \frac{-1}{(x-3)^2}$
• (b) $f'(x) = \frac{-1}{(x+2)^2}$

Would you like more detailed steps for either part, or explanations on a specific step?

Here are 5 related questions for practice:

1. How does the first principles method differ from using derivative rules?
2. Can you find the derivative of $f(x) = \frac{1}{x+1}$ using first principles?
3. What is the derivative of $f(x) = \frac{1}{x^2 + 1}$ by first principles?
4. How would you approach finding the derivative of a quadratic function by first principles?
5. Can you explain the concept of limits as used in the first principles definition of the derivative?

Tip: When working with limits in first principles, always aim to simplify the algebra before taking the limit to zero for cleaner results.